Content-Type: multipart/mixed; boundary="-------------0303240745422" This is a multi-part message in MIME format. ---------------0303240745422 Content-Type: text/plain; name="03-134.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-134.keywords" Hyperbolic KAM theory, flow-box coordinates, Poincar\'e--Melnikov method ---------------0303240745422 Content-Type: application/postscript; name="ms1.pss" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ms1.pss" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: ms1.dvi %%Pages: 36 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips ms1 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.12.19:1440 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 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y(in)n(tegrable)e(Hamiltonian)h (systems)g(with)h Fq(n)14 b Fs(+)f(1)25 b(degrees)f(of)h(freedom,)h Fq(n)14 b Fs(+)f(1)23 b Fp(\025)f Fs(2,)k(the)f(r^)-42 b(ole)25 b(of)g(the)h(in)n(v)-5 b(arian)n(t)24 b(ob)5 b(jects)25 b(is)g(pla)n(y)n(ed)f(b)n(y)-260 4146 y(the)d(whisk)n(ered)f (tori,)i(of)e(dimension)h(at)g(most)f Fq(n)p Fs(,)i(whic)n(h)f(are)f (asso)r(ciated)f(to)i(the)g(resonances)e(of)h(the)i(frequencies)e(of)h (the)g(unp)r(erturb)r(ed)-260 4246 y(in)n(tegrable)j(Hamiltonian.)35 b(An)26 b(expansion)d(in)i(the)g(p)r(erturbation)f(parameter,)g(sa)n(y) g Fq(")p Fs(,)h(rev)n(eals)e(that)i(b)r(oth)g(the)g(unstable)g(and)f (stable)-260 4345 y(manifolds)h(\(the)g(whisk)n(ers\))e(of)i(a)f(whisk) n(ered)g(torus)f(coincide)i(up)g(to)f(an)n(y)g(\014nite)h(order)e(in)i Fq(")p Fs(,)g(giving)f(rise)g(to)g(separatrices.)34 b(Ho)n(w)n(ev)n (er,)-260 4445 y(in)25 b(general,)f(the)h(whisk)n(ers)f(do)g(not)h (coincide,)g(but)h(their)e(distance)h(turns)g(out)f(to)h(b)r(e)g(exp)r (onen)n(tially)g(small)f(with)h(resp)r(ect)g(to)f Fq(")p Fs(.)36 b(This)-260 4545 y(phenomenon,)24 b(called)f Fo(exp)l(onential)t(ly)28 b(smal)t(l)f(splitting)k Fs(of)24 b(separatrices,)e(tak)n(es)g(place)i(in)f(nearly-in)n(tegrable)e (Hamiltonian)j(systems,)-260 4644 y(and)d(w)n(as)g(\014rst)g(detected)h (b)n(y)f(P)n(oincar)n(\023)-39 b(e)19 b([P)n(oi90)m(,)k Fp(x)p Fs(19])e(b)n(y)g(means)g(of)g(what)h(is)f(no)n(w)n(ada)n(ys)e (kno)n(wn)i(as)g(the)h Fo(Poinc)l(ar)n(\023)-40 b(e{Melnikov)27 b(metho)l(d)p Fs(.)-135 4844 y(Moreo)n(v)n(er,)g(for)i(a)h(nearly-in)n (tegrable)d(Hamiltonian)i(with)h Fq(n)20 b Fs(+)f(1)29 b(degrees)g(of)g(freedom,)h(the)g(computation)g(of)f(the)h(splitting)g (of)-260 4943 y(separatrices)c(is)j(a)f(v)n(ery)f(imp)r(ortan)n(t)h (problem.)39 b(Indeed,)28 b(the)h(existence)f(of)h(splitting)f(with)h Fo(tr)l(ansverse)i(homo)l(clinic)h(orbits)k Fs(implies)-260 5043 y(the)28 b(existence)g(of)g(hetero)r(clinic)f(orbits)g(connecting) h(close)f(enough)g(whisk)n(ered)f(tori)i(\(at)g(least)f(for)g(whisk)n (ered)g(tori)g(whose)g(distance)-260 5142 y(is)d(similar)f(to)g(the)i (splitting)f(distance\),)g(and)g(giv)n(es)f(rise)g(to)g(the)i (transition)e(c)n(hains)g(mec)n(hanism,)h(due)g(to)f(Arnold)h([Arn64)o (],)h(designed)-260 5242 y(to)j(detect,)g(for)f Fq(n)18 b Fs(+)g(1)23 b Fp(\025)f Fs(3,)28 b(the)g(phenomenon)f(of)g (instabilit)n(y)h(called)f Fo(A)n(rnold)j(di\013usion)p Fs(.)-135 5441 y(In)22 b(the)g(exp)r(onen)n(tially)f(small)g(case,)h (the)g(v)-5 b(alidation)22 b(of)f(the)h(P)n(oincar)n(\023)-39 b(e{Melnik)n(o)n(v)17 b(metho)r(d)22 b(for)g(detecting)f(the)i (splitting)f(b)r(ecomes)-260 5541 y(a)31 b(di\016cult)h(problem)f(due)h (to)f(its)h Fo(singular)41 b Fs(c)n(haracter.)46 b(This)31 b(v)-5 b(alidation)31 b(w)n(as)f(\014rst)h(carried)f(out)i(for)f(h)n (yp)r(erb)r(olic)g(p)r(erio)r(dic)g(orbits)-260 5641 y(of)e(Hamiltonian)h(systems)f(with)g(2)g(degrees)f(of)i(freedom)f (\(the)h(case)e Fq(n)e Fs(=)g(1)j(in)g(our)g(setting\))h(and)f(for)g (saddle)g(\014xed)g(p)r(oin)n(ts)g(of)h(area)-260 5740 y(preserving)24 b(maps)i(\(see,)g(for)g(instance,)g([DS97,)g(DR98)o (]\),)h(at)f(least)g(when)g(the)g(p)r(erturbation)g(w)n(as)f(small)h (enough)f(near)g(the)h Fo(c)l(omplex)1845 5989 y Fs(1)p eop %%Page: 2 2 2 1 bop -260 -82 a Fo(singularities)42 b Fs(of)35 b(natural)e (parameterizations)f(of)i(the)h(separatrices.)54 b(In)35 b(fact,)h(Lazutkin)e(w)n(as)f(the)i(\014rst)f(to)g(in)n(tro)r(duce)g (complex)-260 17 y(parameterizations)26 b(of)i(the)g(separatrices,)e (in)i(the)h(case)e(of)h(the)g(standard)f(map)h(\(see)g([Laz84)n(,)g (Gel99)o(]\),)h(although)e(in)h(this)h(case)e(one)-260 117 y(needs)h(to)f(consider)g(a)g(di\013eren)n(t)h(unp)r(erturb)r(ed)g (mo)r(del)f(in)h(order)f(to)g(v)-5 b(alidate)27 b(the)h(metho)r(d.)-135 316 y(In)36 b(more)f(dimensions,)j Fq(n)g Fp(\025)f Fs(2,)h(the)e (arithmetic)g(prop)r(erties)g(of)g(the)g(frequencies)g(of)g(the)h Fq(n)p Fs(-dimensional)e(whisk)n(ered)g(torus,)-260 416 y(together)g(with)h(the)g(quasip)r(erio)r(dicit)n(y)f(prop)r(erties)g (of)g(the)h(splitting,)j(also)34 b(ha)n(v)n(e)h(to)g(b)r(e)h(tak)n(en)f (in)n(to)h(accoun)n(t)f(in)h(the)g(v)-5 b(alidation)-260 516 y(of)35 b(the)g(P)n(oincar)n(\023)-39 b(e{Melnik)n(o)n(v)30 b(appro)n(ximation)i(to)j(the)g(size)f(of)h(the)g(splitting.)58 b(This)35 b(w)n(as)f(detected)h(in)g([Sim94)o(],)i(and)d(rigurously) -260 615 y(established)27 b(in)h([DGJS97])f(for)h(the)g(quasip)r(erio)r (dically)e(forced)h(p)r(endulum.)-135 814 y(On)35 b(the)h(other)f (hand,)i(for)e Fq(n)p Fs(-dimensional)f(whisk)n(ered)h(tori)g(of)g(a)g (Hamiltonian)g(with)h Fq(n)24 b Fs(+)f(1)35 b(degrees)f(of)h(freedom,)i (it)f(turns)-260 914 y(out)28 b([Eli94)o(,)g(DG00)o(])g(that)g(the)g (splitting)g(v)n(ector)f(distance)g(and)h(the)g(Melnik)n(o)n(v)f(v)n (ector)f(function)j(are)d(the)j(gradien)n(t)d(of)i(scalar)e(func-)-260 1014 y(tions,)j(called)f(resp)r(ectiv)n(ely)g(splitting)h(p)r(oten)n (tial)g(and)f(Melnik)n(o)n(v)g(p)r(oten)n(tial.)40 b(This)28 b(prop)r(ert)n(y)-7 b(,)28 b(closely)g(related)g(to)h(the)g(Lagrangian) -260 1113 y(c)n(haracter)36 b(of)h(the)h(whisk)n(ers,)h(implies)e (directly)h(the)f(existence)h(of)f(homo)r(clinic)g(in)n(tersections)g (and)g(translates)f(the)i(problem)f(of)-260 1213 y(searc)n(hing)28 b(for)i(suc)n(h)f(in)n(tersections)g(to)h(the)g(problem)g(of)g(searc)n (hing)e(for)h(critical)g(p)r(oin)n(ts)h(of)g(a)g(scalar)e(function.)44 b(In)31 b(the)f(same)f(w)n(a)n(y)-7 b(,)-260 1313 y(the)28 b(transv)n(erse)e(in)n(tersections)g(can)h(b)r(e)h(translated)f(to)h (nondegenerate)e(critical)h(p)r(oin)n(ts.)-135 1512 y(Suc)n(h)j(study)h (has)f(b)r(een)i(carried)d(out)i(in)g([Sau01)n(,)g(R)-9 b(W00,)30 b(LMS99],)h(b)n(y)g(lo)r(oking)e(at)i(t)n(w)n(o)f(di\013eren) n(t)h(solutions)f(of)g(a)h(Hamilton{)-260 1611 y(Jacobi)c(equation,)h (whic)n(h)f(corresp)r(ond)g(to)h(b)r(oth)g(whisk)n(ers)f(of)h(the)g (torus.)37 b(Their)28 b(di\013erence)g(is)g(sho)n(wn)f(to)h(b)r(e)g (exp)r(onen)n(tially)g(small)-260 1711 y(in)k(the)g(parameter)f(p)r (erturbation)g Fq(")g Fs(b)n(y)h(expressing)e(it)i(in)h(some)e(\015o)n (w-b)r(o)n(x)f(v)-5 b(ariables)30 b(for)i(the)g Fq(n)21 b Fs(+)g(1)31 b(v)-5 b(ariables)31 b(of)g(the)h(whisk)n(ers,)-260 1811 y(where)27 b(it)h(is)g(a)f(quasip)r(erio)r(dic)g(function.)-135 2010 y(In)36 b(the)g(presen)n(t)f(pap)r(er,)i Fo(\015ow-b)l(ox)g(c)l(o) l(or)l(dinates)44 b Fs(are)34 b(in)n(tro)r(duced)i(in)g(a)f(small)g (neigh)n(b)r(ourho)r(o)r(d)g(con)n(taining)f(a)i(piece)f(of)h(b)r(oth) -260 2110 y(whisk)n(ers)24 b(\(but)j(excluding)e(the)h(whisk)n(ered)f (torus\),)g(straigh)n(tening)g(in)g(this)h(w)n(a)n(y)f(all)g(the)h(2)p Fq(n)14 b Fs(+)g(2)25 b(co)r(ordinates)f(and)i(not)f(only)g(half)h(of) -260 2209 y(them)i(\(see)f(also)f([PV01)o(]\).)37 b(These)26 b(co)r(ordinates)g(are)g(global)g(in)h(the)h(angular)d(v)-5 b(ariables)26 b(of)h(the)h(whisk)n(ers,)e(and)g(w)n(e)h(construct)g (them)-260 2309 y(with)h(the)g(help)g(of)g(an)f(iterativ)n(e)g(pro)r (cess)f(whic)n(h)i(allo)n(ws)e(us)h(to)h(con)n(trol)e(their)i(complex)f (domain.)-135 2508 y(The)k(\015o)n(w-b)r(o)n(x)e(co)r(ordinates)h(pro)n (vide)g(a)h(to)r(ol)f(to)h(study)h(the)f(splitting)h(of)f(the)g(p)r (erturb)r(ed)g(whisk)n(ers)f(in)h(the)h(singular)e(case,)h(as)-260 2608 y(w)n(ell)23 b(as)f(their)h(homo)r(clinic)g(in)n(tersections,)g (using)g(the)g(P)n(oincar)n(\023)-39 b(e{Melnik)n(o)n(v)18 b(metho)r(d.)36 b(Indeed,)24 b(these)g(co)r(ordinates)d(allo)n(w)h(us)h (to)g(tak)n(e)-260 2707 y(adv)-5 b(an)n(tage)26 b(of)i(the)g(quasip)r (erio)r(dicit)n(y)e(prop)r(erties)h(of)h(the)g(splitting,)g(closely)e (related)h(to)h(its)f(exp)r(onen)n(tial)h(smallness.)-135 2907 y(As)i(a)f(direct)h(application)f(of)h(the)g(results)f(obtained,)h (sharp)f Fo(upp)l(er)j(b)l(ounds)37 b Fs(follo)n(w)29 b(for)g(the)h(exp)r(onen)n(tially)g(small)f(splitting)h(of)-260 3006 y(separatrices.)35 b(It)27 b(is)h(w)n(orth)e(noting)h(that)h(the)g (straigh)n(tening)e(of)h(all)g(the)h(2)p Fq(n)17 b Fs(+)g(2)27 b(co)r(ordinates)f(seems)h(to)g(indicate)h(that)f(suc)n(h)g(result)-260 3106 y(is)h(also)e(v)-5 b(alid)28 b(in)g(other)f(non-Hamiltonian)g (settings.)37 b(Therefore,)27 b(w)n(e)g(b)r(eliev)n(e)h(that)g(suc)n(h) f(\015o)n(w-b)r(o)n(x)f(co)r(ordinates)g(can)i(b)r(e)g(useful)g(in)-260 3205 y(other)j(situations)g(where)g(a)g(total)g(description)g(of)h(the) f(dynamics)g(close)g(to)g(a)h(whisk)n(er)e(of)h(a)g(Hamiltonian)h (system)f(is)g(needed,)i(as)-260 3305 y(w)n(ell)28 b(as)e(in)i(other)f (settings,)h(lik)n(e)f(rev)n(ersible)f(systems.)-135 3504 y(Besides,)g(w)n(e)g(pro)n(vide)f(an)h(accurate)g(upp)r(er)g(b)r (ound)h(for)f(the)h(size)f(of)h(the)f(error)f(term)h(of)h(the)g(Melnik) n(o)n(v)e(function.)37 b(In)27 b(the)g(pap)r(er)-260 3604 y([DG02],)g(for)g(some)g(more)g(concrete)f(p)r(erturbations)h(in)g (the)h(case)f(of)g(3)g(degrees)f(of)h(freedom)g(\()p Fq(n)c Fs(=)g(2\),)k(it)h(is)f(seen)h(as)e(a)h(consequence)-260 3704 y(of)c(this)h(b)r(ound)f(that)g(the)h(Melnik)n(o)n(v)e(function)h (dominates)g(the)g(error)f(term.)35 b(This)23 b(implies)g(the)h(v)-5 b(alidit)n(y)23 b(of)g(the)g(P)n(oincar)n(\023)-39 b(e{Melnik)n(o)n(v) -260 3803 y(metho)r(d)26 b(to)f(giv)n(e)f(asymptotic)h(estimates)g(for) g(the)g(splitting,)h(sho)n(wing)e(the)i(existence)f(of)g(transv)n(erse) e(homo)r(clinic)i(orbits.)35 b(This)26 b(can)-260 3903 y(b)r(e)i(done)f(thanks)h(to)f(the)h(fact)g(that)g(all)f(the)h(results) f(obtained)h(in)f(the)h(presen)n(t)f(pap)r(er)g(are)g(quan)n(titativ)n (e)g(enough.)-135 4102 y(A)h(more)e(precise)h(description)g(of)h(the)g (setting)f(and)h(of)f(the)h(results)f(of)h(the)g(presen)n(t)f(pap)r(er) g(follo)n(ws.)-260 4639 y Fn(1.1)112 b(Setup:)51 b(A)37 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b(metho)r(d,)34 b(to)f(b)r(e)f(used)h(in)g(the)g (pap)r(er)-260 5441 y([DG02].)-135 5641 y(In)f(Section)h(2,)h(w)n(e)e (establish)g(the)h(p)r(ersistence)f(of)h(the)g(h)n(yp)r(erb)r(olic)f (torus)g(and)g(its)h(lo)r(cal)f(whisk)n(ers)f(under)i(the)g(p)r (erturbation.)-260 5740 y(T)-7 b(o)32 b(suc)n(h)g(end,)i(w)n(e)e(put)h (our)f(Hamiltonian)g(in)g(lo)r(cal)g Fo(hyp)l(erb)l(olic)37 b(c)l(o)l(or)l(dinates)c Fs(\()p Fq(u;)14 b(v)s(;)g(';)g(I)7 b Fs(\),)34 b(in)f(whic)n(h)f(the)h(unp)r(erturb)r(ed)g(whisk)n(ers) 1845 5989 y(4)p eop %%Page: 5 5 5 4 bop -260 -82 a Fs(b)r(ecome)31 b(co)r(ordinate)e(planes.)46 b(Then,)32 b(w)n(e)e(apply)g(the)h Fo(hyp)l(erb)l(olic)k(KAM)e(the)l (or)l(em)e Fs(\(in)g(the)g(v)n(ersion)e(of)i([Nie00)o(]\),)h(whic)n(h)f (pro)n(vides,)-260 17 y(after)26 b(a)g(symplectic)g(c)n(hange,)g(a)f (normal)h(form)g(for)f(the)i(p)r(erturb)r(ed)f(Hamiltonian,)h(ensuring) e(the)i(p)r(ersistence)f(of)g(the)g(torus)g(and)g(its)-260 117 y(lo)r(cal)31 b(whisk)n(ers.)49 b(It)32 b(is)f(imp)r(ortan)n(t)h (to)f(con)n(trol)g(the)h(loss)f Fq(\016)k Fs(of)d(complex)f(domain)h (in)g(the)g(angles)f Fq(')h Fs(\(this)g(will)g(b)r(e)g(useful)g(in)h (order)-260 217 y(to)f(estimate)f(the)h(exp)r(onen)n(tially)f(small)g (splitting,)i(b)n(y)e(c)n(ho)r(osing)f Fq(\016)j Fs(=)c Fq(")2085 187 y Fl(a)2156 217 y Fs(for)i(some)g Fq(a)e(>)h Fs(0\).)48 b(W)-7 b(e)32 b(also)e(giv)n(e)h(an)g(impro)n(v)n(emen)n(t)f (of)-260 316 y(the)h(estimates)e(v)-5 b(alid)30 b(if)h Fq(h)p Fs(\()p Fq(x)p Fs(\))d(=)e Fp(O)879 328 y Fk(2)917 316 y Fs(\()p Fq(x)p Fs(\))31 b(in)f(\(3\),)h(i.e.)f(when)g(the)h (torus)e Fp(T)2079 328 y Fk(0)2147 316 y Fs(remains)g(\014xed)h(under)g (the)g(p)r(erturbation)g(\(for)f(instance)-260 416 y Fq(h)p Fs(\()p Fq(x)p Fs(\))24 b(=)f(cos)13 b Fq(x)19 b Fp(\000)f Fs(1,)27 b(see)g(also)g([DG02)o(]\).)38 b(The)27 b(impro)n(v)n(emen)n(t)g(consists)f(in)i(smaller)f(exp)r(onen)n(ts)g (of)h Fq(\016)i Fs(in)e(the)g(estimates.)-135 615 y(In)i(Section)g(3,)g 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y(will)k(b)r(e)g(suitable)f(enough)g(for)h (our)e(purp)r(oses.)-135 1711 y(In)32 b(Section)h(4,)g(w)n(e)f(mainly)h (sho)n(w)e(that)i(a)f(piece)g(of)h(b)r(oth)g(p)r(erturb)r(ed)f(whisk)n (ers)f(actually)h(en)n(ter)g(in)h(the)g(domain)f(of)g(the)h(\015o)n(w-) -260 1811 y(b)r(o)n(x)38 b(co)r(ordinates,)i(de\014ne)f(a)f(function)h (measuring)e(the)i(distance)f(b)r(et)n(w)n(een)h(the)f(whisk)n(ers,)i (and)f(compare)e(this)i(function)g(with)-260 1910 y(the)h(Melnik)n(o)n (v)f(function.)74 b(T)-7 b(o)39 b(suc)n(h)h(end,)j(w)n(e)c(\014rst)g (de\014ne)h(parameterizations)e Fq(Z)2507 1880 y Fi(\006)2563 1910 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))40 b(for)f(the)h(lo)r(cal)g(p)r (erturb)r(ed)g(whisk)n(ers)-260 2010 y(\(Section)d(4.1\),)i(whic)n(h)e (can)g(b)r(e)g(extended)g(in)h(a)e(natural)g(w)n(a)n(y)g(to)h(global)f (whisk)n(ers)f(with)j(the)f(help)h(of)f(the)g Fo(extension)h(the)l(or)l (em)-260 2110 y Fs(\(Section)20 b(4.2\).)34 b(Next)20 b(w)n(e)g(de\014ne)g(the)g Fo(Melnikov)25 b(function)20 b Fq(M)9 b Fs(\()p Fq(s;)14 b(\022)r Fs(\),)22 b(and)d(sho)n(w)g(that)i (it)f(is)g(the)g(gradien)n(t)f(of)h(a)f(scalar)f(function)j Fq(L)p Fs(\()p Fq(s;)14 b(\022)r Fs(\),)-260 2209 y(called)27 b(the)h Fo(Melnikov)j(p)l(otential)d Fs(\(Section)f(4.3\).)37 b(W)-7 b(e)27 b(sho)n(w)g(that)g(the)h(Melnik)n(o)n(v)e(function)i(pro) n(vides)d(a)i(\014rst)g(order)f(appro)n(ximation)-260 2309 y(in)31 b Fq(\026)f Fs(for)g(the)g(splitting)h(distance)f (\(Section)g(4.4\).)45 b(Ho)n(w)n(ev)n(er,)29 b(w)n(e)h(ha)n(v)n(e)f (to)h(translate)f(the)i(parameterizations)d(of)i(the)g(whisk)n(ers)f (to)-260 2408 y(the)g(\015o)n(w-b)r(o)n(x)e(co)r(ordinates)h(in)h (order)e(to)h(ha)n(v)n(e)g(a)g Fo(quasip)l(erio)l(dic)k Fs(function)d(giving)f(the)h(splitting)g(distance)f(\(Section)h(4.5\).) 40 b(After)29 b(a)-260 2508 y(further)23 b(reparameterization,)f(w)n(e) h(de\014ne)g(the)h Fo(splitting)i(function)d Fp(M)p Fs(\()p Fq(s;)14 b(\022)r Fs(\),)25 b(whic)n(h)e(b)r(ecomes)g(also)f(a)h 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b(upp)l(er)e(b)l(ounds)f Fs(for)f(the)h(splitting)g (function)g Fp(M)g Fs(restricted)f(to)g(the)h(real)f(domain)g(\(see)h (Theorem)e(12\):)45 b(under)31 b(a)g(condition)-260 3305 y Fq(\026)23 b Fp(\024)g Fq(c)-63 3317 y Fk(1)-26 3305 y Fq(")13 3275 y Fl(p)47 3250 y Fh(\003)86 3305 y Fs(,)28 b(w)n(e)f(pro)n(vide)f(a)h(b)r(ound)h(of)g(the)g(t)n(yp)r(e)1294 3550 y Fp(jMj)23 b(\024)1570 3494 y Fq(c)1606 3506 y Fk(2)1643 3494 y Fq(\026)p 1560 3531 143 4 v 1560 3607 a(")1599 3583 y Fl(p)1633 3567 y Fh(\003\003)1726 3550 y Fs(exp)1867 3409 y Fj(\()1934 3550 y Fp(\000)2019 3489 y Fq(C)2098 3422 y Fj(\000)2146 3456 y Fl(\031)p 2146 3470 41 4 v 2150 3518 a Fk(2)2197 3489 y Fq(;)14 b(\032)2277 3422 y Fj(\001)p 2009 3531 317 4 v 2009 3609 a Fq(")2048 3585 y Fk(1)p Fl(=)p Fk(\(2)p Fl(\034)7 b Fk(+2\))2335 3409 y Fj(\))2415 3550 y Fq(;)1448 b Fs(\(7\))-260 3796 y(with)28 b(a)f(constan)n(t)g(whic)n(h)h(dep)r(ends)g(on)f(the)h (complex)f(widths)h(in)g(the)g(parameters)e Fq(s)p Fs(,)i Fq(\022)r 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Fq(;)g Fs(1)p Fq(;)1275 5608 y Fk(1)p 1275 5622 34 4 v 1275 5669 a Fl(\016)1318 5573 y Fj(\001)1356 5641 y Fs(.)48 b(Ho)n(w)n(ev)n(er,)30 b(this)h(feature)g(do)r(es)g(not)g (allo)n(w)f(us)h(to)g(obtain)f(b)r(etter)i(estimates)f(for)-52 5740 y(the)d(splitting.)1845 5989 y(9)p eop %%Page: 10 10 10 9 bop -158 -82 a Fs(4.)41 b(Another)d(tec)n(hnical)g(remark)f(is)h (that)h(the)g(result)f(of)g([Nie00)o(,)k(Th.)c(2.2])g(is)g(stated)g (using)g(alw)n(a)n(ys)f(the)i(suprem)n(um)f(norm.)-52 17 y(Ho)n(w)n(ev)n(er,)23 b(a)i(close)f(lo)r(ok)g(at)h(the)g(pro)r(of)f (sho)n(ws)g(that)h(the)g(b)r(ounds)g(giv)n(en)g(in)g(that)g(pap)r(er)f (do)h(not)g(c)n(hange)f(with)h(the)g(norm)f(\(11\).)-158 182 y(5.)41 b(It)30 b(can)f(also)g(b)r(e)h(seen)f(from)g(the)h(pro)r (of)g(giv)n(en)e(in)i([Nie00])f(that)h(the)g(third)g(condition)g(of)f (\(26\))h(can)f(b)r(e)h(remo)n(v)n(ed)e(in)i(the)g(case)-52 281 y(of)h(\014xed)h(tori.)48 b(This)31 b(is)h(due)g(to)f(the)h(fact)g (that)f Fq(b)1533 251 y Fi(0)1586 281 y 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Fs(\()1566 1746 y Fl(g)1600 1721 y Fh(\000)p Fe(1)1678 1746 y Fk(\()p Fl(E)s Fk(\))1782 1752 y Fs(\))1814 1746 y Fl(S)1862 1782 y Fq(;)180 b(v)26 b Fs(=)d Fq(v)2259 1794 y Fk(0)2296 1782 y Fq(e)2335 1746 y Fi(\000)p Fl(g)2421 1721 y Fh(0)2444 1752 y Fs(\()2476 1746 y Fl(g)2510 1721 y Fh(\000)p Fe(1)2588 1746 y Fk(\()p Fl(E)s Fk(\))2692 1752 y Fs(\))2724 1746 y Fl(S)2772 1782 y Fq(:)-260 2010 y Fs(Let)29 b Fp(j)p Fs(Re)14 b Fq(S)5 b Fp(j)24 b(\024)g Fq(\024)p Fs(,)k Fp(j)p Fs(Im)14 b Fq(S)5 b Fp(j)25 b(\024)f Fq(\033)s Fs(,)29 b Fp(j)p Fq(E)5 b Fp(j)24 b(\024)g Fq(\021)s Fs(,)29 b(and)f(w)n(e)h(w)n(an)n(t) e(to)i(see)f(that)g Fp(j)p Fq(u)p Fp(j)c(\024)g Fq(\015)2225 2022 y Fk(1)2263 2010 y Fs(,)k Fp(j)p Fq(v)s Fp(j)d(\024)f Fq(r)2554 2022 y Fk(1)2592 2010 y Fs(.)39 b(Since)29 b Fq(g)s Fs(\(0\))24 b(=)g(0)k(and)g Fq(g)3409 1980 y Fi(0)3432 2010 y Fs(\(0\))c(=)g(1,)29 b(w)n(e)f(can)-260 2109 y(assume)f(\(taking)g Fq(\021)k Fs(small)c(enough\))h(that,)g(for) f Fp(j)p Fq(E)5 b Fp(j)23 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Fk(\(0\))1001 3823 y Fl(k)1090 3798 y Fs(\()p Fq(S;)14 b(E)5 b(;)14 b(I)7 b Fs(\))23 b(=)g Fq(Q)1565 3810 y Fl(k)1619 3681 y Fj(\022)1680 3798 y Fq(S;)14 b(E)24 b Fp(\000)1964 3742 y Fs(1)p 1946 3779 78 4 v 1946 3855 a(2)p Fq(b)2047 3798 y Fp(h)p Fs(\003)p Fq(I)7 b(;)14 b(I)7 b Fp(i)14 b Fq(;)g(I)2386 3681 y Fj(\023)2461 3798 y Fq(e)2500 3764 y Fl(i)2533 3741 y Ff(S)p 2533 3750 38 4 v 2539 3784 a(b)2580 3764 y Fi(h)p Fl(k)q(;)p Fk(\003)p Fl(I)5 b Fi(i)2774 3798 y Fq(;)-260 4020 y Fs(and)28 b(w)n(e)f(deduce)h(from)f(\(39\))g(that)1252 4049 y Fj(\014)1252 4099 y(\014)1252 4148 y(\014)1279 4144 y Fq(Q)1345 4101 y Fk(\(0\))1345 4169 y Fl(k)1434 4049 y Fj(\014)1434 4099 y(\014)1434 4148 y(\014)1462 4202 y Fl(\024;\033)o(;\021)1612 4210 y Fe(1)1644 4202 y Fl(;\014)s(\016)1763 4144 y Fp(\024)c(j)p Fq(Q)1940 4156 y Fl(k)1981 4144 y Fp(j)2004 4169 y Fl(\024;\033)o(;\021)r(;\015) 2211 4177 y Fe(1)2265 4144 y Fp(\001)18 b Fq(e)2345 4110 y Fi(j)p Fl(k)q Fi(j)p Fl(\016)2458 4144 y Fq(;)-260 4323 y Fs(and)28 b(then)716 4327 y Fj(\015)716 4377 y(\015)716 4427 y(\015)763 4423 y Fq(Q)829 4388 y Fk(\()o(0\))917 4327 y Fj(\015)917 4377 y(\015)917 4427 y(\015)963 4481 y Fl(\024;\033)o(;\021)1113 4489 y Fe(1)1145 4481 y Fl(;\032)1199 4489 y Fe(3)1232 4481 y Fl(;\014)s(\016)1352 4423 y Fp(\024)1460 4344 y Fj(X)1439 4523 y Fl(k)q Fi(2)p Fg(Z)1566 4506 y Ff(n)1614 4423 y Fp(j)p Fq(Q)1703 4435 y Fl(k)1744 4423 y Fp(j)1767 4448 y Fl(\024;\033)o(;\021)r(;\015)1974 4456 y Fe(1)2028 4423 y Fp(\001)18 b Fq(e)2108 4388 y Fi(j)p Fl(k)q Fi(j)p Fk(\()p Fl(\016)r Fk(+)p Fl(\032)2327 4396 y Fe(3)2360 4388 y Fk(\))2413 4423 y Fs(=)23 b Fp(k)p Fq(Q)p Fp(k)2650 4448 y Fl(\024;\033)o(;\021)r(;\032)2856 4456 y Fe(2)2888 4448 y Fl(;\015)2943 4456 y Fe(1)2993 4423 y Fq(:)3930 4624 y Fd(2)-260 4823 y Ft(Remarks.)-158 5080 y Fs(1.)41 b(T)-7 b(o)29 b(ha)n(v)n(e)g(a)g(reduction)g Fq(\016)k Fs(of)d(the)g(complex)g(domain)f(in)h(the)g(angles)f Fq(')p Fs(,)i(w)n(e)e(ha)n(v)n(e)g(accepted)g(a)g(drastic)g(reduction)h (of)g(domain)-52 5180 y(in)d(the)g(actions)f Fq(I)7 b Fs(.)37 b(Ho)n(w)n(ev)n(er,)25 b(this)i(is)f(not)h(imp)r(ortan)n(t)g(b) r(ecause)f(for)g Fq(\026)h Fs(small)g(enough)f(the)h(reduced)f(domain)h (still)g(con)n(tains)f(a)-52 5280 y(neigh)n(b)r(ourho)r(o)r(d)g(of)i(b) r(oth)g(whisk)n(ers,)e(as)h(sho)n(wn)g(in)h(Prop)r(osition)e(9.)-158 5441 y(2.)41 b(According)34 b(to)g(this)i(prop)r(osition,)f(an)g (\\aniso)r(c)n(hronous")d(system)i(with)i(equations)2727 5420 y(_)2704 5441 y Fq(S)k Fs(=)35 b Fq(b)p Fs(,)59 b(_)-46 b Fq(')36 b Fs(=)41 b(~)-48 b Fq(!)3232 5453 y Fl(")3290 5441 y Fs(+)23 b(\003)p Fq(I)42 b Fs(can)34 b(b)r(e)i(tak)n(en)-52 5541 y(in)n(to)30 b(an)g(\\iso)r(c)n(hronous")e (system)1074 5520 y(_)1050 5541 y Fq(S)33 b Fs(=)28 b Fq(b)p Fs(,)54 b(_)-46 b Fq(')29 b Fs(=)34 b(~)-49 b Fq(!)1544 5553 y Fl(")1579 5541 y Fs(.)46 b(A)n(t)31 b(a)g(\014rst)f(view,)h(this)g(fact)g(can)f(seem)h(strange,)f(but)h(it) g(is)g(due)f(to)h(that)-52 5641 y(the)d(\015o)n(w-b)r(o)n(x)f(co)r (ordinates)g(are)g(de\014ned)i(for)e(\()p Fq(S;)14 b(')p Fs(\))29 b(in)g(\(a)f(subset)g(of)6 b(\))29 b(the)g(cylinder)f Fm(R)c Fp(\002)19 b Fm(T)2922 5610 y Fl(n)2966 5641 y Fs(,)29 b(and)f(the)g(in)n(v)-5 b(arian)n(t)28 b(torus)f(\(at)-52 5740 y Fq(S)g Fs(=)c Fp(\0061)p Fs(\))28 b(has)f(b)r(een)h(excluded)f (from)h(the)g(domain)f(of)g(these)h(co)r(ordinates.)1825 5989 y(12)p eop %%Page: 13 13 13 12 bop -260 -82 a Ft(Theorem)31 b(4)g(\(\015o)m(w-b)s(o)m(x)h (theorem\))40 b Fo(L)l(et)29 b Fq(K)1287 -112 y Fk(\(0\))1406 -82 y Fo(as)h(in)g(\(36\),)h(analytic)g(on)e Fp(B)2309 -70 y Fl(\024;\033)o(;\021)2459 -62 y Fe(1)2491 -70 y Fl(;\032)2545 -62 y Fe(3)2577 -70 y Fl(;\014)s(\016)2674 -82 y Fo(.)39 b(Assume)29 b(that)1454 6 y Fj(\015)1454 56 y(\015)1454 106 y(\015)1501 101 y Fq(Q)1567 67 y Fk(\()o(0\))1655 6 y Fj(\015)1655 56 y(\015)1655 106 y(\015)1702 160 y Fl(\024;\033)o(;\021)1852 168 y Fe(1)1883 160 y Fl(;\032)1937 168 y Fe(3)1970 160 y Fl(;\014)s(\016)2090 101 y Fp(\026)22 b Fq(\016)2217 67 y Fk(2)2255 101 y Fq(:)1567 b Fs(\(40\))-260 298 y Fo(Then,)31 b(ther)l(e)f(exists)f(a)h(symple)l(ctic)h(map)g Fs(\007)22 b(:)h Fp(B)1258 310 y Fl(\024)1297 318 y Fe(1)1329 310 y Fl(;\033)1387 318 y Fe(1)1420 310 y Fl(;\021)1474 318 y Fe(2)1506 310 y Fl(;\032)1560 318 y Fe(4)1593 310 y Fl(;\014)1651 318 y Fe(1)1683 310 y Fl(\016)1742 298 y Fp(\000)-14 b(!)23 b(B)1954 310 y Fl(\024;\033)o(;\021)2104 318 y Fe(1)2135 310 y Fl(;\032)2189 318 y Fe(3)2222 310 y Fl(;\014)s(\016)2319 298 y Fo(,)30 b(with)g Fs(\007)18 b Fp(\000)g Fs(id)23 b(=)g Fp(O)r Fs(\()p Fq(E)5 b(;)14 b(I)7 b Fs(\))p Fo(,)31 b(such)f(that)1591 464 y Fq(K)1668 429 y Fk(\()o(0\))1775 464 y Fp(\016)18 b Fs(\007)23 b(=)f Fq(K)2081 476 y Fk(0)2118 464 y Fq(;)-260 619 y Fo(and)30 b(the)g(fol)t(lowing)j(b)l(ound)c(holds:)954 820 y Fp(j)p Fs(\007)18 b Fp(\000)g Fs(id)p Fp(j)1235 845 y Fl(\024)1274 853 y Fe(1)1307 845 y Fl(;\033)1365 853 y Fe(1)1397 845 y Fl(;\021)1451 853 y Fe(2)1484 845 y Fl(;\032)1538 853 y Fe(4)1570 845 y Fl(;\014)1628 853 y Fe(1)1660 845 y Fl(\016)1719 820 y Fp(\026)1807 725 y Fj(\015)1807 775 y(\015)1807 824 y(\015)1853 820 y Fq(Q)1919 786 y Fk(\(0\))2008 725 y Fj(\015)2008 775 y(\015)2008 824 y(\015)2054 878 y Fl(\024;\033)o(;\021)2204 886 y Fe(1)2236 878 y Fl(;\032)2290 886 y Fe(3)2323 878 y Fl(;\014)s(\016)2438 820 y Fp(\001)2479 703 y Fj(\022)2540 820 y Fs(1)p Fq(;)2629 764 y Fs(1)p 2629 801 42 4 v 2630 877 a Fq(\016)2680 703 y Fj(\023)2755 820 y Fq(:)1067 b Fs(\(41\))-135 1121 y(The)33 b(pro)r(of)g(of)g(this)g(result)g(in)n (v)n(olv)n(es)f(a)h(standard)f(con)n(v)n(ergen)n(t)f(iterativ)n(e)h (pro)r(cess,)i(based)e(in)i(Lie)f(series,)h(and)f(w)n(e)g(defer)g(it)g (to)-260 1220 y(App)r(endix)c(B.)-260 1420 y Ft(Remark.)54 b Fs(W)-7 b(e)28 b(see)f(from)h(\(37\),)f(\(34\))g(and)h(\(31\))f(that) 842 1505 y Fj(\015)842 1555 y(\015)842 1605 y(\015)888 1600 y Fq(Q)954 1566 y Fk(\(0\))1043 1505 y Fj(\015)1043 1555 y(\015)1043 1605 y(\015)1089 1658 y Fl(\024;\033)o(;\021)1239 1666 y Fe(1)1271 1658 y Fl(;\032)1325 1666 y Fe(3)1358 1658 y Fl(;\014)s(\016)1478 1600 y Fp(\024)22 b(k)p Fq(Q)p Fp(k)1714 1625 y Fl(\024;\033)o(;\021)r(;\032)1920 1633 y Fe(2)1952 1625 y Fl(;\015)2007 1633 y Fe(1)2066 1600 y Fp(\024)h(k)o Fq(R)q Fp(k)2301 1625 y Fl(r)2332 1633 y Fe(1)2364 1625 y Fl(;\015)2419 1633 y Fe(1)2451 1625 y Fl(;\032)2505 1633 y Fe(2)2565 1600 y Fp(\026)2735 1544 y Fq(\026)p 2662 1581 195 4 v 2662 1657 a(\016)2702 1633 y Fl(p)2736 1641 y Fe(3)2769 1633 y Fk(+1)2867 1600 y Fq(;)955 b Fs(\(42\))-260 1796 y(and)28 b(hence)f(condition)h(\(40\)) f(can)g(b)r(e)h(written)g(as)1677 1896 y Fq(\026)23 b Fp(\026)g Fq(\016)1878 1862 y Fl(p)1912 1870 y Fe(3)1944 1862 y Fk(+3)2032 1896 y Fq(:)1790 b Fs(\(43\))-260 2266 y Fr(4)135 b(The)44 b(p)t(erturb)t(ed)g(whisk)l(ers)i(and)f(the)g (splitting)h(function)-260 2564 y Fn(4.1)112 b(Lo)s(cal)38 b(whisk)m(ers)-260 2817 y Fs(First,)44 b(w)n(e)c(translate)g(the)g 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3483 y Fp(W)847 3448 y Fk(+)840 3505 y(0)925 3483 y Fs(:)189 b Fq(Y)1204 3448 y Fk(+)1185 3505 y(0)1259 3483 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))24 b(=)f(\(0)p Fq(;)14 b(Ae)1764 3449 y Fi(\000)p Fl(s)1851 3483 y Fq(;)g(\022)r(;)g Fs(0\))p Fq(;)96 b Fs(Re)14 b Fq(s)23 b Fp(\025)g Fq(q)2458 3495 y Fk(0)2495 3483 y Fq(;)42 b Fp(j)p Fs(Im)14 b Fq(\022)r Fp(j)23 b(\024)g Fq(\032)2914 3495 y Fk(1)2951 3483 y Fq(;)871 b Fs(\(45\))-260 3639 y(where)26 b(the)h(constan)n(t)f Fq(A)d(>)g Fs(0)j(is)g(suc)n(h)g (that)h(\()p Fq(x)1222 3651 y Fk(0)1260 3639 y Fs(\()p Fq(s)p Fs(\))p Fq(;)14 b(y)1441 3651 y Fk(0)1478 3639 y Fs(\()p Fq(s)p Fs(\)\))24 b(=)f(\000)1777 3651 y Fk(0)1814 3639 y Fs(\(0)p Fq(;)14 b(Ae)2026 3609 y Fi(\000)p Fl(s)2113 3639 y Fs(\),)27 b(and)f(hence)h Fq(Z)2642 3651 y Fk(0)2679 3639 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))23 b(=)g(\000)3037 3572 y Fj(\000)3075 3639 y Fq(Y)3142 3603 y Fk(+)3123 3661 y(0)3197 3639 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))3378 3572 y Fj(\001)3416 3639 y Fs(.)37 b(In)27 b(fact,)f(w)n(e)h(see)-260 3751 y(from)32 b(\(22\))f(that)i Fq(A)d Fs(=)g(4)534 3683 y Fp(p)p 603 3683 42 4 v 68 x Fs(2)o(.)50 b(W)-7 b(e)32 b(can)g(also)f(tak)n(e)g Fq(q)1413 3763 y Fk(0)1480 3751 y Fs(=)f(ln\()p Fq(A=r)r Fs(\))k(\(and)e(then)g(one)g(has)f Fq(Y)2652 3716 y Fk(+)2633 3774 y(0)2707 3751 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))31 b Fp(2)f(S)3054 3763 y Fl(r)n(;\015)t(;\032) 3196 3771 y Fe(1)3232 3751 y Fs(,)k(the)e(domain)f(of)h Fq(G)3899 3763 y Fk(0)3937 3751 y Fs(\).)-260 3851 y(Analogously)-7 b(,)26 b(w)n(e)h(parameterize)f(the)i(unp)r(erturb)r(ed)h(unstable)e (whisk)n(er)g(b)n(y)763 4007 y Fp(W)852 3971 y Fi(\000)845 4029 y Fk(0)930 4007 y Fs(:)190 b Fq(Y)1209 3971 y Fi(\000)1191 4029 y Fk(0)1265 4007 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))24 b(=)f(\()p Fq(Ae)1691 3973 y Fl(s)1726 4007 y Fq(;)14 b Fs(0)p Fq(;)g(\022)r(;)g Fs(0\))p Fq(;)97 b Fs(Re)13 b Fq(s)23 b Fp(\024)g(\000)p Fq(q)2477 4019 y Fk(0)2514 4007 y Fq(;)41 b Fp(j)q Fs(Im)14 b Fq(\022)r Fp(j)23 b(\024)g 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Fs(\(48\))-260 5288 y Fn(4.2)112 b(Global)37 b(whisk)m(ers)g(and)h(the)g(extension)f(theorem)-260 5541 y Fs(No)n(w,)26 b(w)n(e)f(extend)g(the)h(parameterizations)e(of)h (the)h(whisk)n(ers)e Fp(W)1805 5505 y Fi(\006)1798 5566 y Fk(lo)r(c)1885 5541 y Fs(,)i(v)-5 b(alid)26 b(for)f Fp(\006)14 b Fs(Re)f Fq(s)23 b Fp(\025)g Fq(q)2635 5553 y Fk(2)2672 5541 y Fs(,)j(to)g(further)f(v)-5 b(alues)25 b(of)h Fq(s)f Fs(using)g(the)h(fact)-260 5641 y(that)32 b(the)g(whisk)n(ers)e(are)g(formed)h(b)n(y)g(tra)5 b(jectories)30 b(of)h(our)f(Hamiltonian)i Fq(H)7 b Fs(.)47 b(W)-7 b(e)32 b(denote)f Fp(W)2777 5610 y Fi(\006)2865 5641 y Fs(the)h Fo(glob)l(al)i(whiskers)f Fs(obtained)e(in)-260 5740 y(this)d(w)n(a)n(y)-7 b(.)1825 5989 y(14)p eop %%Page: 15 15 15 14 bop -135 -82 a Fs(The)38 b(extension)h(theorem)f(giv)n(en)g(b)r (elo)n(w)g(pro)n(vides)f(a)i(b)r(ound)g(for)f(the)h(distance)f(b)r(et)n (w)n(een)h(the)g(global)f(p)r(erturb)r(ed)h(whisk)n(ers)-260 17 y(and)32 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b(=)g Fq(@)2669 5480 y Fl(\022)2707 5468 y Fq(L)p Fs(\()p Fq(s;)14 b(\022)r Fs(\),)27 b(where)555 5699 y Fq(L)p Fs(\()p Fq(s;)14 b(\022)r Fs(\))24 b(=)e Fp(\000)983 5586 y Fj(Z)1066 5607 y Fi(1)1029 5775 y(\0001)1165 5699 y Fs([)p Fq(H)1257 5711 y Fk(1)1308 5699 y Fs(\()p Fq(Z)1397 5711 y Fk(0)1448 5699 y Fs(\()p Fq(s)d Fs(+)f Fq(bt;)c(\022)20 b Fs(+)k(~)-48 b Fq(!)1918 5711 y Fl(")1953 5699 y Fq(t)p Fs(\)\))19 b Fp(\000)f Fq(H)2218 5711 y Fk(1)2269 5699 y Fs(\()q Fq(Z)2365 5665 y Fi(\003)2359 5720 y Fk(0)2416 5699 y Fs(\()p Fq(\022)j Fs(+)k(~)-49 b Fq(!)2643 5711 y Fl(")2678 5699 y Fq(t)p Fs(\))q(\)])14 b(d)p Fq(t)k Fs(+)g(const)1825 5989 y(15)p eop %%Page: 16 16 16 15 bop -260 -82 a Fs(\(the)27 b(constan)n(t)e(can)h(b)r(e)h(c)n (hosen)e(in)h(suc)n(h)g(a)g(w)n(a)n(y)f(that)h Fq(L)g Fs(has)f(zero)g(a)n(v)n(erage)f(with)i(resp)r(ect)g(to)g Fq(\022)r Fs(;)h(this)g(a)n(v)n(erage)c(do)r(es)j(not)g(dep)r(end)g(on) -260 17 y Fq(s)p Fs(\).)44 b(The)30 b(Melnik)n(o)n(v)f(p)r(oten)n(tial) h 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Fq(t)k Fs(+)g(const)p Fq(;)667 2122 y(M)9 b Fs(\()p Fq(s;)14 b(\022)r Fs(\))84 b(=)e Fp(\000)1248 2009 y Fj(Z)1331 2029 y Fi(1)1294 2197 y(\0001)1430 2122 y Fs([)p Fq(h)p Fs(\()p Fq(x)1580 2134 y Fk(0)1618 2122 y Fs(\()p Fq(s)18 b Fs(+)g Fq(bt)p Fs(\)\))h Fp(\000)f Fq(h)p Fs(\(0\)])h Fp(\001)f Fq(@)2303 2134 y Fl(')2351 2122 y Fq(f)k Fs(\()q Fq(\022)e Fs(+)25 b(~)-49 b Fq(!)2641 2134 y Fl(")2677 2122 y Fq(t)p Fs(\))14 b(d)p Fq(t:)993 b Fs(\(52\))-260 2357 y(Next)28 b(w)n(e)f(pro)n(vide)g (an)g(upp)r(er)h(b)r(ound)g(for)f(the)h(Melnik)n(o)n(v)e(function,)j (to)e(b)r(e)h(used)g(later.)-260 2644 y Ft(Lemma)i(7)41 b Fo(F)-6 b(or)30 b(any)g Fs(\()p Fq(s;)14 b(\022)r Fs(\))30 b Fo(with)h Fp(j)p Fs(Im)14 b Fq(s)p Fp(j)23 b(\024)f Fq(\027)1223 2656 y Fk(1)1261 2644 y Fo(,)30 b Fp(j)p Fs(Im)14 b Fq(\022)r Fp(j)23 b(\024)g Fq(\032)1670 2656 y Fk(2)1707 2644 y Fo(,)30 b(one)g(has)h(in)f(\(52\))g(the)g(b)l(ound)g Fp(j)p Fq(M)9 b Fs(\()p Fq(s;)14 b(\022)r Fs(\))p Fp(j)23 b(\026)g Fs(1)p Fq(=\016)3282 2614 y Fk(2)p Fl(l)p Fk(+)p Fl(\013)3434 2644 y Fo(.)-260 2931 y(Pr)l(o)l(of.)48 b Fs(W)-7 b(e)24 b(\014rst)f(use)g(h)n(yp)r(othesis)f(\(H3\),)j(whic)n (h)e(sa)n(ys)f(that)h Fq(h)p Fs(\()p Fq(x)p Fs(\))h(is)g(a)e (trigonometric)g(p)r(olynomial)g(of)i(degree)e Fq(l)i Fs(and)f(hence)h Fq(h)p Fs(\()p Fq(x)3819 2943 y Fk(0)3856 2931 y Fs(\()p Fq(s)p Fs(\)\))-260 3031 y(has)33 b(p)r(oles)f(of)h (order)f(2)p Fq(l)i Fs(at)e Fq(s)g Fs(=)g Fp(\006)p Fq(i\031)s(=)p Fs(2)f(\(this)j(comes)e(from)h(the)g(equalit)n(y)f Fq(e)2226 3001 y Fl(ix)2287 3009 y Fe(0)2320 3001 y Fk(\()p Fl(s)p Fk(\))2439 3031 y Fs(=)g(\()p Fq(i)21 b Fp(\000)h Fs(sinh)14 b Fq(s)p Fs(\))2938 3001 y Fk(2)2975 3031 y Fq(=)g Fs(cosh)3188 2994 y Fk(2)3239 3031 y Fq(s)p Fs(\).)53 b(This)33 b(implies,)i(for) -260 3131 y Fp(j)p Fs(Im)14 b Fq(s)p Fp(j)23 b(\024)g Fq(\027)90 3143 y Fk(1)150 3131 y Fs(=)g Fq(\031)s(=)p Fs(2)18 b Fp(\000)g Fq(\016)s Fs(,)27 b(the)h(b)r(ound)1177 3169 y Fj(Z)1260 3190 y Fi(1)1223 3358 y(\0001)1359 3282 y Fp(j)p Fq(h)p Fs(\()p Fq(x)1509 3294 y Fk(0)1547 3282 y Fs(\()p Fq(s)19 b Fs(+)f Fq(bt)p Fs(\)\))g Fp(\000)g Fq(h)p Fs(\(0\))p Fp(j)c Fs(d)p Fq(t)23 b Fp(\026)2410 3226 y Fs(1)p 2339 3263 184 4 v 2339 3339 a Fq(\016)2379 3315 y Fk(2)p Fl(l)p Fi(\000)p Fk(1)2532 3282 y Fq(;)-260 3485 y Fs(whic)n(h)30 b(can)f(b)r(e)g(deduced)h(as)f(a)g(consequence)f (of)i(Lemma)f(15)f(in)i(App)r(endix)g(A.)43 b(On)29 b(the)h(other)f (hand,)h(w)n(e)f(deduce)g(from)h(h)n(yp)r(othe-)-260 3585 y(sis)d(\(H4\))h(that)g Fp(j)p Fq(@)299 3597 y Fl(')347 3585 y Fq(f)9 b Fp(j)420 3613 y Fl(\032)454 3621 y Fe(2)514 3585 y Fp(\026)23 b Fs(1)p Fq(=\016)726 3555 y Fl(\013)p Fk(+1)856 3585 y Fs(,)28 b(and)f(w)n(e)g(then)i(obtain)e(the)h(exp)r (ected)g(b)r(ound)g(for)f Fp(j)p Fq(M)9 b Fs(\()p Fq(s;)14 b(\022)r Fs(\))p Fp(j)p Fs(.)1080 b Fd(2)-260 3924 y Fn(4.4)112 b(A)37 b(\014rst)h(order)f(appro)m(ximation)f(for)i(the)f (splitting)-260 4179 y Fs(In)30 b(this)g(section)g(w)n(e)f(sho)n(w)g (that,)i(for)f Fq(\026)g Fs(small)f(enough,)h(the)g(Melnik)n(o)n(v)f (function)i Fq(M)38 b Fs(giv)n(es)29 b(a)g(\014rst)h(order)e(appro)n (ximation)g(for)i(the)-260 4278 y(splitting)36 b(b)r(et)n(w)n(een)g (the)g(global)f(whisk)n(ers)f Fp(W)1239 4248 y Fi(\006)1331 4278 y Fs(\(in)i(the)g(original)f(co)r(ordinates\))f(pro)n(viding,)j (in)f(addition,)h(an)f(estimate)g(for)f(the)-260 4378 y(remainder)27 b(\(often)h(called)f(the)h(\\error)d(term"\))j(in)f (terms)h(of)f(the)h(loss)f Fq(\016)k Fs(of)c(complex)g(domain.)-135 4577 y(Recall)32 b(that,)i(in)f(Theorem)f(6,)i(the)f(parameterizations) d Fq(Z)1776 4547 y Fk(+)1831 4577 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))33 b(and)g Fq(Z)2275 4547 y Fi(\000)2331 4577 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))33 b(ha)n(v)n(e)f(b)r(een)h (extended)g(to)f(Re)14 b Fq(s)32 b Fp(\025)f(\000)p Fq(q)3788 4589 y Fi(\003)3858 4577 y Fs(and)-260 4677 y(Re)14 b Fq(s)24 b Fp(\024)f Fq(q)40 4689 y Fi(\003)106 4677 y Fs(resp)r(ectiv)n(ely)-7 b(.)38 b(Hence)28 b(their)g(splitting)g(can)g (b)r(e)g(measured)g(for)f Fp(j)p Fs(Re)14 b Fq(s)p Fp(j)24 b(\024)f Fq(q)2470 4689 y Fi(\003)2508 4677 y Fs(.)39 b(In)28 b(fact,)g(w)n(e)g(measure)f(the)h(splitting)h(in)f(the)-260 4776 y(\()p Fq(n)p Fs(-dimensional\))f Fq(I)7 b Fs(-direction,)27 b(considering)f(the)i Fq(I)7 b Fs(-comp)r(onen)n(t)27 b(of)h(the)g(parameterizations:)35 b Fq(Z)2859 4741 y Fi(\000)2853 4801 y Fl(I)2914 4776 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))19 b Fp(\000)f Fq(Z)3260 4741 y Fk(+)3254 4801 y Fl(I)3315 4776 y Fs(\()p Fq(s;)c(\022)r Fs(\).)-260 5064 y Ft(Prop)s(osition)30 b(8)42 b Fo(Consider)f(the)f(p)l(ar)l (ameterizations)i Fq(Z)1599 5033 y Fi(\006)1655 5064 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))40 b Fo(as)h(in)f(The)l(or)l(em)h(6,) i(and)e(assume)f(c)l(onditions)h(\(26\))f(and)h(\(49\).)-260 5163 y(Then,)31 b(for)1067 5268 y Fp(j)q Fs(Re)13 b Fq(s)p Fp(j)23 b(\024)g Fq(q)1412 5280 y Fi(\003)1450 5268 y Fq(;)184 b Fp(j)p Fs(Im)14 b Fq(s)p Fp(j)23 b(\024)g Fq(\027)2007 5280 y Fk(1)2044 5268 y Fq(;)184 b Fp(j)p Fs(Im)14 b Fq(\022)r Fp(j)23 b(\024)g Fq(\032)2605 5280 y Fk(3)2642 5268 y Fq(;)-260 5422 y Fo(and)30 b(with)h(the)f(exp)l (onents)f Fq(p)642 5434 y Fl(i)699 5422 y Fo(de\014ne)l(d)h(in)g (\(14{16\),)i(one)e(has)h(the)e(b)l(ound)751 5591 y Fj(\014)751 5641 y(\014)779 5661 y Fq(Z)842 5626 y Fi(\000)836 5686 y Fl(I)897 5661 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))19 b Fp(\000)f Fq(Z)1243 5626 y Fk(+)1237 5686 y Fl(I)1298 5661 y Fs(\()p Fq(s;)c(\022)r Fs(\))19 b Fp(\000)f Fq(\026M)9 b Fs(\()p Fq(s;)14 b(\022)r Fs(\))1902 5591 y Fj(\014)1902 5641 y(\014)1953 5661 y Fp(\026)2168 5605 y Fs(1)p 2051 5642 277 4 v 2051 5718 a Fq(\016)2091 5694 y Fk(2)p Fl(l)p Fk(+)p Fl(\013)p Fk(+1)2351 5544 y Fj(\022)2434 5605 y Fq(\026)2484 5575 y Fk(2)p 2423 5642 111 4 v 2423 5718 a Fq(\016)2463 5694 y Fl(p)2497 5702 y Fe(4)2562 5661 y Fs(+)2720 5605 y Fq(\026)2770 5575 y Fk(2)p 2655 5642 219 4 v 2655 5719 a Fq(\016)2695 5695 y Fl(p)2729 5703 y Fe(2)2766 5659 y Fp(p)p 2835 5659 39 4 v 60 x Fq(")2883 5544 y Fj(\023)2958 5661 y Fq(:)1825 5989 y Fs(16)p eop %%Page: 17 17 17 16 bop -260 -82 a Fo(Pr)l(o)l(of.)57 b Fs(T)-7 b(o)27 b(b)r(egin,)h(w)n(e)f(write)714 101 y Fq(Z)777 66 y Fi(\000)771 126 y Fl(I)833 101 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))19 b Fp(\000)f Fq(Z)1179 66 y Fk(+)1173 126 y Fl(I)1233 101 y Fs(\()p Fq(s;)c(\022)r Fs(\))24 b(=)1526 34 y Fj(\002)1560 101 y Fq(Z)1623 66 y Fi(\000)1617 126 y Fl(I)1679 101 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))19 b Fp(\000)f Fq(Z)2025 67 y Fi(\003)2019 122 y Fl(I)2063 101 y Fs(\()p Fq(\022)r Fs(\))2168 34 y Fj(\003)2222 101 y Fp(\000)2305 34 y Fj(\002)2339 101 y Fq(Z)2402 66 y Fk(+)2396 126 y Fl(I)2457 101 y Fs(\()p Fq(s;)c(\022)r Fs(\))19 b Fp(\000)f Fq(Z)2803 67 y Fi(\003)2797 122 y Fl(I)2841 101 y Fs(\()p Fq(\022)r Fs(\))2946 34 y Fj(\003)2995 101 y Fq(:)-260 299 y Fs(Using)28 b(that)173 278 y(_)153 299 y Fq(I)i Fs(=)23 b Fp(\000)p Fq(\026@)466 311 y Fl(')514 299 y Fq(H)583 311 y Fk(1)648 299 y Fs(is)k(one)g(of)h(the)g(Hamiltonian)f(equations)g(asso)r(ciated) f(to)i Fq(H)h Fs(=)23 b Fq(H)2722 311 y Fk(0)2778 299 y Fs(+)18 b Fq(\026H)2980 311 y Fk(1)3017 299 y Fs(,)28 b(w)n(e)f(get)713 533 y Fq(Z)776 497 y Fk(+)770 557 y Fl(I)831 533 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))19 b Fp(\000)f Fq(Z)1177 498 y Fi(\003)1171 553 y Fl(I)1215 533 y Fs(\()p Fq(\022)r Fs(\))24 b(=)e Fp(\000)1510 420 y Fj(Z)1593 440 y Fi(1)1556 608 y Fk(0)1702 477 y Fs(d)p 1687 514 77 4 v 1687 590 a(d)p Fq(t)1787 465 y Fj(\002)1821 533 y Fq(Z)1884 497 y Fk(+)1878 557 y Fl(I)1953 533 y Fs(\()p Fq(s)c Fs(+)g Fq(bt;)c(\022)21 b Fs(+)j(~)-48 b Fq(!)2423 545 y Fl(")2458 533 y Fq(t)p Fs(\))19 b Fp(\000)f Fq(Z)2685 498 y Fi(\003)2679 553 y Fl(I)2736 533 y Fs(\()q Fq(\022)i Fs(+)25 b(~)-49 b Fq(!)2963 545 y Fl(")2999 533 y Fq(t)p Fs(\))3061 465 y Fj(\003)3109 533 y Fs(d)p Fq(t)879 759 y Fs(=)23 b Fq(\026)1031 646 y Fj(Z)1114 667 y Fi(1)1077 835 y Fk(0)1198 692 y Fj(\002)1232 759 y Fq(@)1276 771 y Fl(')1324 759 y Fq(H)1393 771 y Fk(1)1444 692 y Fj(\000)1482 759 y Fq(Z)1545 725 y Fk(+)1614 759 y Fs(\()p Fq(s)c Fs(+)f Fq(bt;)c(\022)20 b Fs(+)k(~)-48 b Fq(!)2084 771 y Fl(")2119 759 y Fq(t)p Fs(\))2181 692 y Fj(\001)2238 759 y Fp(\000)18 b Fq(@)2365 771 y Fl(')2413 759 y Fq(H)2482 771 y Fk(1)2533 759 y Fs(\()p Fq(Z)2628 725 y Fi(\003)2680 759 y Fs(\()p Fq(\022)j Fs(+)j(~)-48 b Fq(!)2907 771 y Fl(")2942 759 y Fq(t)p Fs(\)\))3037 692 y Fj(\003)3085 759 y Fs(d)p Fq(t:)-260 999 y Fs(Pro)r(ceeding)26 b(similarly)g(with)h Fq(Z)758 963 y Fi(\000)752 1023 y Fl(I)814 999 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))j Fp(\000)g Fq(Z)1157 968 y Fi(\003)1151 1021 y Fl(I)1195 999 y Fs(\()p Fq(\022)r Fs(\),)28 b(and)f(taking)f(in) n(to)h(accoun)n(t)f(the)h(expression)e(\(50\))i(of)g(the)g(Melnik)n(o)n (v)f(function,)i(w)n(e)-260 1098 y(obtain)516 1282 y Fq(Z)579 1246 y Fi(\000)573 1306 y Fl(I)635 1282 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))19 b Fp(\000)f Fq(Z)981 1246 y Fk(+)975 1306 y Fl(I)1036 1282 y Fs(\()p Fq(s;)c(\022)r Fs(\))19 b Fp(\000)f Fq(\026M)9 b Fs(\()p Fq(s;)14 b(\022)r Fs(\))599 1469 y(=)83 b Fp(\000)p Fq(\026)876 1352 y Fj(\024)919 1356 y(Z)1002 1377 y Fk(0)965 1545 y Fi(\0001)1101 1402 y Fj(\002)1136 1469 y Fq(@)1180 1481 y Fl(')1228 1469 y Fq(H)1297 1481 y Fk(1)1348 1402 y Fj(\000)1386 1469 y Fq(Z)1449 1435 y Fi(\000)1505 1402 y Fj(\001)1561 1469 y Fp(\000)18 b Fq(@)1688 1481 y Fl(')1736 1469 y Fq(H)1805 1481 y Fk(1)1856 1469 y Fs(\()p Fq(Z)1945 1481 y Fk(0)1982 1469 y Fs(\))h Fp(\000)f Fq(@)2160 1481 y Fl(')2208 1469 y Fq(H)2277 1481 y Fk(1)2328 1469 y Fs(\()p Fq(Z)2423 1435 y Fi(\003)2461 1469 y Fs(\))h(+)f Fq(@)2639 1481 y Fl(')2687 1469 y Fq(H)2756 1481 y Fk(1)2807 1469 y Fs(\()p Fq(Z)2902 1435 y Fi(\003)2896 1490 y Fk(0)2940 1469 y Fs(\))2972 1402 y Fj(\003)3021 1469 y Fs(d)p Fq(t)932 1705 y Fs(+)1011 1592 y Fj(Z)1094 1613 y Fi(1)1057 1781 y Fk(0)1178 1638 y Fj(\002)1212 1705 y Fq(@)1256 1717 y Fl(')1304 1705 y Fq(H)1373 1717 y Fk(1)1424 1638 y Fj(\000)1462 1705 y Fq(Z)1525 1671 y Fk(+)1580 1638 y Fj(\001)1637 1705 y Fp(\000)g Fq(@)1764 1717 y Fl(')1811 1705 y Fq(H)1880 1717 y Fk(1)1932 1705 y Fs(\()p Fq(Z)2021 1717 y Fk(0)2058 1705 y Fs(\))h Fp(\000)f Fq(@)2236 1717 y Fl(')2283 1705 y Fq(H)2352 1717 y Fk(1)2404 1705 y Fs(\()p Fq(Z)2499 1671 y Fi(\003)2537 1705 y Fs(\))g(+)g Fq(@)2714 1717 y Fl(')2762 1705 y Fq(H)2831 1717 y Fk(1)2882 1705 y Fs(\()q Fq(Z)2978 1671 y Fi(\003)2972 1726 y Fk(0)3015 1705 y Fs(\))3048 1638 y Fj(\003)3096 1705 y Fs(d)p Fq(t)3172 1588 y Fj(\025)3845 1705 y Fs(\(53\))-260 1934 y(where,)27 b(for)g(shortness,)g(w)n(e)g(denote)1067 2118 y Fq(Z)1130 2084 y Fi(\006)1209 2118 y Fs(=)22 b Fq(Z)1359 2084 y Fi(\006)1429 2118 y Fs(\()p Fq(s)c Fs(+)h Fq(bt;)14 b(\022)20 b Fs(+)k(~)-48 b Fq(!)1899 2130 y Fl(")1934 2118 y Fq(t)p Fs(\))14 b Fq(;)97 b(Z)2193 2084 y Fi(\003)2254 2118 y Fs(=)22 b Fq(Z)2404 2084 y Fi(\003)2456 2118 y Fs(\()p Fq(\022)f Fs(+)k(~)-49 b Fq(!)2683 2130 y Fl(")2718 2118 y Fq(t)p Fs(\))14 b Fq(;)1028 b Fs(\(54\))1067 2242 y Fq(Z)1124 2254 y Fk(0)1184 2242 y Fs(=)23 b Fq(Z)1329 2254 y Fk(0)1379 2242 y Fs(\()q Fq(s)18 b Fs(+)g Fq(bt;)c(\022)20 b Fs(+)25 b(~)-49 b Fq(!)1849 2254 y Fl(")1885 2242 y Fq(t)p Fs(\))14 b Fq(;)97 b(Z)2144 2208 y Fi(\003)2138 2263 y Fk(0)2204 2242 y Fs(=)23 b Fq(Z)2355 2208 y Fi(\003)2349 2263 y Fk(0)2407 2242 y Fs(\()p Fq(\022)e Fs(+)j(~)-48 b Fq(!)2634 2254 y Fl(")2669 2242 y Fq(t)p Fs(\))14 b Fq(:)1077 b Fs(\(55\))-260 2426 y(Since)28 b(the)g(t)n(w)n(o)e(in)n (tegrals)g(in)i(\(53\))f(are)f(analogous,)g(w)n(e)h(are)f(going)g(to)i (b)r(ound)g(only)f(one)g(of)g(them,)h(sa)n(y)e(the)i(second)f(one.)36 b(W)-7 b(e)28 b(break)-260 2526 y(the)g(in)n(tegral)185 2459 y Fj(R)241 2479 y Fi(1)225 2555 y Fk(0)339 2526 y Fs(in)g(three)f(parts,)g Fp(I)929 2538 y Fk(1)985 2526 y Fs(+)18 b Fp(I)1113 2538 y Fk(2)1169 2526 y Fs(+)g Fp(I)1297 2538 y Fk(3)1334 2526 y Fs(,)28 b(de\014ned)g(as)f(follo)n (ws,)g(with)h(a)f(suitable)g Fq(t)2666 2538 y Fk(0)2704 2526 y Fs(\()p Fq(s)p Fs(\):)741 2784 y Fp(I)786 2796 y Fk(1)847 2784 y Fs(=)c Fp(\000)1014 2671 y Fj(Z)1096 2691 y Fl(t)1121 2699 y Fe(0)1153 2691 y Fk(\()p Fl(s)p Fk(\))1059 2859 y(0)1255 2784 y Fs([)p Fq(@)1322 2796 y Fl(')1370 2784 y Fq(H)1439 2796 y Fk(1)1490 2784 y Fs(\()p Fq(Z)1585 2749 y Fi(\003)1623 2784 y Fs(\))c Fp(\000)f Fq(@)1801 2796 y Fl(')1848 2784 y Fq(H)1917 2796 y Fk(1)1969 2784 y Fs(\()p Fq(Z)2064 2749 y Fi(\003)2058 2804 y Fk(0)2102 2784 y Fs(\)])c(d)p Fq(t;)741 3028 y Fp(I)786 3040 y Fk(2)847 3028 y Fs(=)935 2915 y Fj(Z)1018 2936 y Fl(t)1043 2944 y Fe(0)1075 2936 y Fk(\()p Fl(s)p Fk(\))981 3104 y(0)1176 2961 y Fj(\002)1211 3028 y Fq(@)1255 3040 y Fl(')1303 3028 y Fq(H)1372 3040 y Fk(1)1423 2961 y Fj(\000)1461 3028 y Fq(Z)1524 2994 y Fk(+)1579 2961 y Fj(\001)1635 3028 y Fp(\000)k Fq(@)1762 3040 y Fl(')1810 3028 y Fq(H)1879 3040 y Fk(1)1930 3028 y Fs(\()p Fq(Z)2019 3040 y Fk(0)2056 3028 y Fs(\))2089 2961 y Fj(\003)2137 3028 y Fs(d)p Fq(t;)741 3255 y Fp(I)786 3267 y Fk(3)847 3255 y Fs(=)935 3142 y Fj(Z)1018 3162 y Fi(1)981 3330 y Fl(t)1006 3338 y Fe(0)1038 3330 y Fk(\()p Fl(s)p Fk(\))1139 3188 y Fj(\002)1174 3255 y Fq(@)1218 3267 y Fl(')1266 3255 y Fq(H)1335 3267 y Fk(1)1386 3188 y Fj(\000)1424 3255 y Fq(Z)1487 3221 y Fk(+)1542 3188 y Fj(\001)1598 3255 y Fp(\000)g Fq(@)1725 3267 y Fl(')1773 3255 y Fq(H)1842 3267 y Fk(1)1893 3255 y Fs(\()q Fq(Z)1983 3267 y Fk(0)2019 3255 y Fs(\))h Fp(\000)f Fq(@)2197 3267 y Fl(')2245 3255 y Fq(H)2314 3267 y Fk(1)2365 3255 y Fs(\()p Fq(Z)2460 3221 y Fi(\003)2498 3255 y Fs(\))h(+)f Fq(@)2676 3267 y Fl(')2724 3255 y Fq(H)2793 3267 y Fk(1)2844 3255 y Fs(\()p Fq(Z)2939 3221 y Fi(\003)2933 3275 y Fk(0)2977 3255 y Fs(\))3009 3188 y Fj(\003)3058 3255 y Fs(d)p Fq(t;)-260 3495 y Fs(and)23 b(our)f(aim)g(is)h(to)g(apply)f(the)h(b)r(ounds)g(of)g (Lemma)f(5)h(to)f(the)h(in)n(tegrals)f Fp(I)2073 3507 y Fk(1)2133 3495 y Fs(and)h Fp(I)2335 3507 y Fk(3)2372 3495 y Fs(,)h(and)f(Theorem)e(6)i(to)f(the)i(in)n(tegral)d Fp(I)3563 3507 y Fk(2)3601 3495 y Fs(.)35 b(Recalling)-260 3595 y(that)28 b Fp(j)p Fs(Re)14 b Fq(s)p Fp(j)23 b(\024)g Fq(q)265 3607 y Fi(\003)331 3595 y Fs(and)k(that)h(w)n(e)f(consider)g Fq(q)1155 3607 y Fi(\003)1216 3595 y Fq(>)c(q)1341 3607 y Fk(2)1378 3595 y Fs(,)28 b(w)n(e)f(c)n(ho)r(ose)1141 3828 y Fq(t)1171 3840 y Fk(0)1208 3828 y Fs(\()p Fq(s)p Fs(\))d(=)1422 3711 y Fj(\032)1526 3777 y Fs(\()p Fq(q)1595 3789 y Fk(2)1651 3777 y Fp(\000)18 b Fs(Re)c Fq(s)p Fs(\))p Fq(=b)110 b Fs(if)28 b(Re)14 b Fq(s)23 b(<)g(q)2480 3789 y Fk(2)2517 3777 y Fs(,)1526 3877 y(0)537 b(if)28 b(Re)14 b Fq(s)23 b Fp(\025)g Fq(q)2480 3889 y Fk(2)2517 3877 y Fs(,)-260 4061 y(so)k(that,)h(in)g(the)g(in)n(tegral)e Fp(I)632 4073 y Fk(3)670 4061 y Fs(,)h(w)n(e)h(ha)n(v)n(e)e(Re)14 b Fq(s)k Fs(+)g Fq(bt)23 b Fp(\024)g Fq(q)1500 4073 y Fk(2)1537 4061 y Fs(.)37 b(Note)27 b(that)h Fq(t)2007 4073 y Fk(0)2045 4061 y Fs(\()p Fq(s)p Fs(\))23 b Fp(\024)g Fs(\()p Fq(q)2328 4073 y Fk(2)2384 4061 y Fs(+)18 b Fq(q)2504 4073 y Fi(\003)2542 4061 y Fs(\))p Fq(=b)23 b Fp(\026)f Fs(1.)-135 4261 y(Among)29 b(the)i(three)f(in)n(tegrals,)f(the)h (second)g(one)f(will)h(ha)n(v)n(e)f(the)i(largest)d(b)r(ound,)j(whic)n (h)f(comes)g(from)f(a)h(direct)g(application)f(of)-260 4360 y(part)e(\(b\))i(of)e(Theorem)g(6:)1276 4511 y Fp(jI)1344 4523 y Fk(2)1382 4511 y Fp(j)c(\026)1643 4455 y Fs(1)p 1525 4492 277 4 v 1525 4568 a Fq(\016)1565 4544 y Fk(2)p Fl(l)p Fk(+)p Fl(\013)p Fk(+1)1826 4394 y Fj(\022)1928 4455 y Fq(\026)p 1897 4492 111 4 v 1897 4568 a(\016)1937 4544 y Fl(p)1971 4552 y Fe(4)2036 4511 y Fs(+)2214 4455 y Fq(\026)p 2129 4492 219 4 v 2129 4569 a(\016)2169 4545 y Fl(p)2203 4553 y Fe(2)2240 4509 y Fp(p)p 2309 4509 39 4 v 60 x Fq(")2358 4394 y Fj(\023)2433 4511 y Fq(:)-135 4812 y Fs(T)-7 b(o)26 b(b)r(ound)h(the)g(in)n(tegrals)e Fp(I)762 4824 y Fk(1)827 4812 y Fs(and)h Fp(I)1032 4824 y Fk(3)1070 4812 y Fs(,)h(w)n(e)f(\014rst)h(write)f(them)h(in)g(terms)g (of)f(the)i(parameterizations)c(`)p Fq(Y)19 b Fs(')27 b(in)n(tro)r(duced)f(in)h(\(44{45\))-260 4911 y(and)h(\(47\):)982 5160 y Fp(I)1027 5172 y Fk(1)1088 5160 y Fs(=)22 b Fp(\000)1254 5047 y Fj(Z)1337 5068 y Fl(t)1362 5076 y Fe(0)1394 5068 y Fk(\()p Fl(s)p Fk(\))1300 5236 y(0)1495 5160 y Fs([)p Fq(F)k Fs(\()p Fq(Y)1696 5126 y Fi(\003)1734 5160 y Fs(\))19 b Fp(\000)f Fq(F)26 b Fs(\()p Fq(Y)2046 5126 y Fi(\003)2027 5181 y Fk(0)2084 5160 y Fs(\)])14 b(d)p Fq(t;)982 5387 y Fp(I)1027 5399 y Fk(3)1088 5387 y Fs(=)1175 5274 y Fj(Z)1258 5294 y Fi(1)1221 5462 y Fl(t)1246 5470 y Fe(0)1279 5462 y Fk(\()p Fl(s)p Fk(\))1380 5319 y Fj(\002)1415 5387 y Fq(F)1493 5319 y Fj(\000)1531 5387 y Fq(Y)1598 5352 y Fk(+)1653 5319 y Fj(\001)1710 5387 y Fp(\000)k Fq(F)1872 5319 y Fj(\000)1910 5387 y Fq(Y)1976 5351 y Fk(+)1958 5409 y(0)2031 5319 y Fj(\001)2088 5387 y Fp(\000)g Fq(F)26 b Fs(\()p Fq(Y)2349 5352 y Fi(\003)2387 5387 y Fs(\))18 b(+)g Fq(F)26 b Fs(\()q Fq(Y)2698 5352 y Fi(\003)2680 5407 y Fk(0)2736 5387 y Fs(\))2769 5319 y Fj(\003)2817 5387 y Fs(d)p Fq(t;)-260 5641 y Fs(where)k(w)n(e)g(denote)h Fq(F)40 b Fs(=)27 b Fq(@)608 5653 y Fl(')656 5641 y Fq(H)725 5653 y Fk(1)783 5641 y Fp(\016)20 b Fs(\000)28 b(=)f Fq(@)1061 5653 y Fl(')1109 5641 y Fq(G)1174 5653 y Fk(1)1242 5641 y Fs(\(see)k(Section)f(2.1\),)h(and)f Fq(Y)2129 5610 y Fk(+)2184 5641 y Fs(,)h Fq(Y)2305 5605 y Fk(+)2286 5663 y(0)2360 5641 y Fs(,)h Fq(Y)2481 5610 y Fi(\003)2519 5641 y Fs(,)g Fq(Y)2641 5610 y Fi(\003)2622 5661 y Fk(0)2709 5641 y Fs(are)e(analogous)e(to)j(\(54{55\).)43 b(W)-7 b(e)31 b(ha)n(v)n(e)-260 5740 y Fq(Y)-193 5705 y Fk(+)-212 5762 y(0)-138 5740 y Fq(;)14 b(Y)-35 5710 y Fi(\003)-53 5761 y Fk(0)27 5740 y Fp(2)23 b(S)155 5752 y Fl(r)186 5760 y Fe(2)219 5752 y Fl(;\015)274 5760 y Fe(2)306 5752 y Fl(;\032)360 5760 y Fe(3)423 5740 y Fs(and,)k(since)g(b)r(ounds)g (\(b\))g(and)g(\(c\))h(of)e(Lemma)h(5)f(are)g Fp(\026)d Fq(\016)s Fs(,)k(w)n(e)g(also)e(ha)n(v)n(e)h Fq(Y)2843 5710 y Fk(+)2898 5740 y Fq(;)14 b(Y)3001 5710 y Fi(\003)3063 5740 y Fp(2)23 b(S)3191 5752 y Fl(r)3222 5760 y Fe(1)3255 5752 y Fl(;\015)3310 5760 y Fe(1)3342 5752 y Fl(;\032)3396 5760 y Fe(2)3432 5740 y Fs(.)37 b(The)27 b(results)f(of)1825 5989 y(17)p eop %%Page: 18 18 18 17 bop -260 -82 a Fs(Lemma)27 b(5)h(can)f(b)r(e)h(applied)f (replacing)g Fq(s)g Fs(b)n(y)h Fq(s)18 b Fs(+)g Fq(bt)p Fs(,)27 b(and)h(pro)n(vide)e(the)i(follo)n(wing)f(three)g(b)r(ounds:) 873 127 y Fp(j)p Fq(Y)963 92 y Fi(\003)1019 127 y Fp(\000)18 b Fq(Y)1169 92 y Fi(\003)1150 147 y Fk(0)1207 127 y Fp(j)23 b(\026)1381 70 y Fq(\026)p 1351 107 111 4 v 1351 184 a(\016)1391 160 y Fl(p)1425 168 y Fe(3)1490 127 y Fp(\001)1532 10 y Fj(\022)1593 127 y Fs(1)p Fq(;)1681 70 y Fs(1)p 1681 107 42 4 v 1682 184 a Fq(\016)1733 10 y Fj(\023)1808 127 y Fq(;)2011 56 y Fj(\014)2011 106 y(\014)2038 127 y Fq(Y)2105 92 y Fk(+)2179 127 y Fp(\000)18 b Fq(Y)2328 91 y Fk(+)2310 149 y(0)2383 56 y Fj(\014)2383 106 y(\014)2434 127 y Fp(\026)2562 70 y Fq(\026)p 2532 107 111 4 v 2532 184 a(\016)2572 160 y Fl(p)2606 168 y Fe(3)2671 127 y Fp(\001)2712 10 y Fj(\022)2773 127 y Fs(1)p Fq(;)2862 70 y Fs(1)p 2862 107 42 4 v 2863 184 a Fq(\016)2913 10 y Fj(\023)2988 127 y Fq(;)873 298 y Fj(\014)873 348 y(\014)901 369 y Fq(Y)967 334 y Fk(+)1041 369 y Fp(\000)g Fq(Y)1191 333 y Fk(+)1172 391 y(0)1264 369 y Fp(\000)g Fq(Y)1414 334 y Fi(\003)1471 369 y Fs(+)g Fq(Y)1620 334 y Fi(\003)1602 389 y Fk(0)1658 298 y Fj(\014)1658 348 y(\014)1709 369 y Fp(\026)1807 312 y Fq(e)1846 282 y Fi(\000)p Fk(\(Re)10 b Fl(s)p Fk(+)p Fl(bt)p Fk(\))2179 312 y Fq(\026)p 1807 349 423 4 v 1962 425 a(\016)2002 402 y Fl(p)2036 410 y Fe(3)2257 369 y Fp(\001)2299 251 y Fj(\022)2360 369 y Fs(1)p Fq(;)2448 312 y Fs(1)p 2448 349 42 4 v 2449 425 a Fq(\016)2500 251 y Fj(\023)2575 369 y Fq(:)-260 582 y Fs(Note)28 b(also)e(from)i(\(44{45\))d(that)1292 611 y Fj(\014)1292 661 y(\014)1320 682 y Fq(Y)1387 646 y Fk(+)1368 704 y(0)1460 682 y Fp(\000)18 b Fq(Y)1610 647 y Fi(\003)1591 702 y Fk(0)1648 611 y Fj(\014)1648 661 y(\014)1699 682 y Fp(\026)k Fq(e)1825 647 y Fi(\000)p Fk(\(Re)11 b Fl(s)p Fk(+)p Fl(bt)p Fk(\))2177 682 y Fp(\001)19 b Fs(\(1)p Fq(;)14 b Fs(0\))f Fq(:)-260 820 y Fs(W)-7 b(e)28 b(are)f(also)f(going)h(to)g(use)h(the)g(next)f(b)r(ounds,)h (easily)f(deduced)h(from)f(\(25\):)545 1033 y Fp(j)p Fs(D)p Fq(F)12 b Fp(j)719 1058 y Fl(r)750 1066 y Fe(1)783 1058 y Fl(;\015)838 1066 y Fe(1)870 1058 y Fl(;\032)924 1066 y Fe(2)983 1033 y Fp(\026)1146 977 y Fs(1)p 1081 1014 172 4 v 1081 1090 a Fq(\016)1121 1066 y Fl(\013)p Fk(+1)1281 1033 y Fp(\001)1322 916 y Fj(\022)1383 1033 y Fs(1)p Fq(;)1472 977 y Fs(1)p 1472 1014 42 4 v 1473 1090 a Fq(\016)1523 916 y Fj(\023)1598 1033 y Fq(;)1801 962 y Fj(\014)1801 1012 y(\014)1829 1033 y Fs(D)1892 999 y Fk(2)1929 1033 y Fq(F)1994 962 y Fj(\014)1994 1012 y(\014)2022 1066 y Fl(r)2053 1074 y Fe(1)2085 1066 y Fl(;\015)2140 1074 y Fe(1)2172 1066 y Fl(;\032)2226 1074 y Fe(2)2286 1033 y Fp(\026)2449 977 y Fs(1)p 2384 1014 172 4 v 2384 1090 a Fq(\016)2424 1066 y Fl(\013)p Fk(+1)2579 916 y Fj(\022)2722 982 y Fs(1)142 b(1)p Fq(=\016)2681 1082 y Fs(1)p Fq(=\016)86 b Fs(1)p Fq(=\016)3012 1052 y Fk(2)3089 916 y Fj(\023)3164 1033 y Fq(;)658 b Fs(\(56\))-260 1246 y(where)40 b(the)h(elemen)n(ts)f(of)g(the)h(matrix)f(giv)n(e)f(b)r (ounds)i(for)f(the)h(second)e(partial)h(deriv)-5 b(ativ)n(es)39 b(\(analogously)f(to)j(the)f(notation)g(of)-260 1346 y(Section)28 b(1.5\).)-135 1545 y(Using)f(these)h(ingredien)n(ts,)e (the)i(in)n(tegral)f Fp(I)1247 1557 y Fk(1)1312 1545 y Fs(is)g(easily)g(b)r(ounded:)1063 1729 y Fp(jI)1131 1741 y Fk(1)1168 1729 y Fp(j)c(\026)1302 1637 y Fj(D)1353 1729 y Fp(j)p Fs(D)p Fq(F)12 b Fp(j)1527 1754 y Fl(r)1558 1762 y Fe(1)1590 1754 y Fl(;\015)1645 1762 y Fe(1)1677 1754 y Fl(;\032)1731 1762 y Fe(2)1782 1729 y Fq(;)i Fp(j)p Fq(Y)1909 1695 y Fi(\003)1965 1729 y Fp(\000)k Fq(Y)2115 1695 y Fi(\003)2096 1750 y Fk(0)2153 1729 y Fp(j)2176 1637 y Fj(E)2250 1729 y Fp(\026)2467 1673 y Fq(\026)p 2348 1710 290 4 v 2348 1786 a(\016)2388 1762 y Fl(p)2422 1770 y Fe(3)2454 1762 y Fk(+)p Fl(\013)p Fk(+3)2646 1729 y Fq(:)-260 1916 y Fs(The)28 b(in)n(tegral)e Fp(I)258 1928 y Fk(3)323 1916 y Fs(requires)h(some)g(extra)f(w)n(ork.)36 b(First,)27 b(w)n(e)g(write)455 2080 y Fq(F)534 2013 y Fj(\000)572 2080 y Fq(Y)639 2046 y Fk(+)694 2013 y Fj(\001)750 2080 y Fp(\000)18 b Fq(F)912 2013 y Fj(\000)950 2080 y Fq(Y)1017 2044 y Fk(+)998 2102 y(0)1072 2013 y Fj(\001)1128 2080 y Fp(\000)g Fq(F)26 b Fs(\()p Fq(Y)1389 2046 y Fi(\003)1427 2080 y Fs(\))19 b(+)f Fq(F)26 b Fs(\()p Fq(Y)1739 2046 y Fi(\003)1720 2100 y Fk(0)1777 2080 y Fs(\))538 2272 y(=)686 2159 y Fj(Z)769 2180 y Fk(1)732 2348 y(0)820 2205 y Fj(\012)859 2272 y Fs(D)p Fq(F)1001 2205 y Fj(\000)1039 2272 y Fq(Y)1106 2236 y Fk(+)1087 2294 y(0)1180 2272 y Fs(+)18 b Fq(u)1325 2205 y Fj(\000)1362 2272 y Fq(Y)1429 2238 y Fk(+)1502 2272 y Fp(\000)g Fq(Y)1652 2236 y Fk(+)1633 2294 y(0)1707 2205 y Fj(\001\001)1797 2272 y Fq(;)c(Y)1900 2238 y Fk(+)1974 2272 y Fp(\000)k Fq(Y)2124 2236 y Fk(+)2105 2294 y(0)2179 2205 y Fj(\013)2232 2272 y Fs(d)p Fq(u)686 2511 y Fp(\000)765 2398 y Fj(Z)847 2418 y Fk(1)810 2586 y(0)898 2511 y Fp(h)q Fs(D)p Fq(F)26 b Fs(\()p Fq(Y)1172 2476 y Fi(\003)1153 2531 y Fk(0)1228 2511 y Fs(+)18 b Fq(u)c Fs(\()p Fq(Y)1472 2476 y Fi(\003)1528 2511 y Fp(\000)k Fq(Y)1678 2476 y Fi(\003)1659 2531 y Fk(0)1716 2511 y Fs(\)\))d Fq(;)f(Y)1898 2476 y Fi(\003)1955 2511 y Fp(\000)k Fq(Y)2104 2476 y Fi(\003)2086 2531 y Fk(0)2143 2511 y Fp(i)c Fs(d)p Fq(u)538 2750 y Fs(=)686 2637 y Fj(Z)769 2657 y Fk(1)732 2825 y(0)820 2682 y Fj(\012)859 2750 y Fs(D)p Fq(F)1001 2682 y Fj(\000)1039 2750 y Fq(Y)1106 2714 y Fk(+)1087 2772 y(0)1180 2750 y Fs(+)k Fq(u)1325 2682 y Fj(\000)1362 2750 y Fq(Y)1429 2715 y Fk(+)1502 2750 y Fp(\000)g Fq(Y)1652 2714 y Fk(+)1633 2772 y(0)1707 2682 y Fj(\001\001)1802 2750 y Fp(\000)g Fs(D)p Fq(F)26 b Fs(\()p Fq(Y)2126 2715 y Fi(\003)2107 2770 y Fk(0)2182 2750 y Fs(+)18 b Fq(u)c Fs(\()p Fq(Y)2426 2715 y Fi(\003)2482 2750 y Fp(\000)k Fq(Y)2632 2715 y Fi(\003)2613 2770 y Fk(0)2670 2750 y Fs(\)\))c Fq(;)g(Y)2852 2715 y Fk(+)2926 2750 y Fp(\000)k Fq(Y)3075 2714 y Fk(+)3057 2772 y(0)3130 2682 y Fj(\013)3183 2750 y Fs(d)p Fq(u)686 2988 y Fs(+)765 2875 y Fj(Z)847 2896 y Fk(1)810 3064 y(0)898 2921 y Fj(\012)938 2988 y Fs(D)p Fq(F)26 b Fs(\()p Fq(Y)1179 2954 y Fi(\003)1160 3009 y Fk(0)1235 2988 y Fs(+)18 b Fq(u)c Fs(\()p Fq(Y)1479 2954 y Fi(\003)1535 2988 y Fp(\000)k Fq(Y)1685 2954 y Fi(\003)1666 3009 y Fk(0)1723 2988 y Fs(\)\))c Fq(;)g(Y)1905 2954 y Fk(+)1979 2988 y Fp(\000)k Fq(Y)2128 2953 y Fk(+)2110 3010 y(0)2202 2988 y Fp(\000)g Fq(Y)2351 2954 y Fi(\003)2408 2988 y Fs(+)g Fq(Y)2558 2954 y Fi(\003)2539 3009 y Fk(0)2596 2921 y Fj(\013)2649 2988 y Fs(d)p Fq(u:)-260 3194 y Fs(Then,)28 b(using)f(the)h(previous)f(b)r(ounds)g(w)n(e)h(obtain:)520 3287 y Fj(\014)520 3337 y(\014)547 3358 y Fq(F)626 3290 y Fj(\000)664 3358 y Fq(Y)731 3323 y Fk(+)786 3290 y Fj(\001)843 3358 y Fp(\000)18 b Fq(F)1004 3290 y Fj(\000)1042 3358 y Fq(Y)1109 3322 y Fk(+)1090 3380 y(0)1164 3290 y Fj(\001)1221 3358 y Fp(\000)g Fq(F)25 b Fs(\()q Fq(Y)1481 3323 y Fi(\003)1519 3358 y Fs(\))19 b(+)f Fq(F)26 b Fs(\()p Fq(Y)1831 3323 y Fi(\003)1812 3378 y Fk(0)1869 3358 y Fs(\))1901 3287 y Fj(\014)1901 3337 y(\014)603 3515 y Fp(\026)750 3423 y Fj(D)801 3445 y(\014)801 3495 y(\014)829 3515 y Fs(D)892 3481 y Fk(2)929 3515 y Fq(F)994 3445 y Fj(\014)994 3495 y(\014)1022 3548 y Fl(r)1053 3556 y Fe(1)1085 3548 y Fl(;\015)1140 3556 y Fe(1)1172 3548 y Fl(;\032)1226 3556 y Fe(2)1281 3515 y Fp(\001)1323 3448 y Fj(\000)1361 3445 y(\014)1361 3495 y(\014)1389 3515 y Fq(Y)1455 3480 y Fk(+)1437 3537 y(0)1529 3515 y Fp(\000)18 b Fq(Y)1679 3481 y Fi(\003)1660 3536 y Fk(0)1717 3445 y Fj(\014)1717 3495 y(\014)1763 3515 y Fs(+)1846 3445 y Fj(\014)1846 3495 y(\014)1874 3515 y Fq(Y)1940 3481 y Fk(+)2014 3515 y Fp(\000)g Fq(Y)2163 3480 y Fk(+)2145 3537 y(0)2237 3515 y Fp(\000)g Fq(Y)2387 3481 y Fi(\003)2443 3515 y Fs(+)g Fq(Y)2593 3481 y Fi(\003)2574 3536 y Fk(0)2631 3445 y Fj(\014)2631 3495 y(\014)2659 3448 y(\001)2711 3515 y Fq(;)2748 3445 y Fj(\014)2748 3495 y(\014)2775 3515 y Fq(Y)2842 3481 y Fk(+)2915 3515 y Fp(\000)g Fq(Y)3065 3480 y Fk(+)3046 3537 y(0)3120 3445 y Fj(\014)3120 3495 y(\014)3148 3423 y(E)750 3698 y Fs(+)829 3606 y Fj(D)880 3698 y Fp(j)p Fs(D)p Fq(F)12 b Fp(j)1054 3723 y Fl(r)1085 3731 y Fe(1)1117 3723 y Fl(;\015)1172 3731 y Fe(1)1204 3723 y Fl(;\032)1258 3731 y Fe(2)1309 3698 y Fq(;)1346 3627 y Fj(\014)1346 3677 y(\014)1373 3698 y Fq(Y)1440 3664 y Fk(+)1514 3698 y Fp(\000)18 b Fq(Y)1663 3662 y Fk(+)1645 3720 y(0)1737 3698 y Fp(\000)g Fq(Y)1886 3664 y Fi(\003)1943 3698 y Fs(+)g Fq(Y)2093 3664 y Fi(\003)2074 3718 y Fk(0)2131 3627 y Fj(\014)2131 3677 y(\014)2159 3606 y(E)603 3915 y Fp(\026)760 3859 y Fq(e)799 3829 y Fi(\000)p Fk(\(Re)11 b Fl(s)p Fk(+)p Fl(bt)p Fk(\))1133 3859 y Fq(\026)p 760 3896 423 4 v 827 3972 a(\016)867 3948 y Fl(p)901 3956 y Fe(3)934 3948 y Fk(+)p Fl(\013)p Fk(+3)1193 3915 y Fq(;)2629 b Fs(\(57\))-260 4102 y(and)28 b(then)1581 4202 y Fp(jI)1649 4214 y Fk(3)1686 4202 y Fp(j)23 b(\026)1949 4145 y Fq(\026)p 1830 4182 290 4 v 1830 4258 a(\016)1870 4234 y Fl(p)1904 4242 y Fe(3)1936 4234 y Fk(+)p Fl(\013)p Fk(+3)2129 4202 y Fq(:)-260 4367 y Fs(Note)h(that)h(it)g(is)f(essen)n(tial)f(to)i(ha)n(v)n(e)e(the)h (exp)r(onen)n(tial)g(factor)g(in)g(\(57\),)h(b)r(ecause)f(this)g(b)r (ound)h(has)f(to)g(b)r(e)h(in)n(tegrated)e(o)n(v)n(er)f(an)i (in\014nite)-260 4467 y(in)n(terv)-5 b(al.)-135 4666 y(No)n(w)27 b(w)n(e)g(compare)f(the)i(b)r(ounds)g(obtained)f(for)g Fp(jI)1476 4678 y Fk(1)1513 4666 y Fp(j)p Fs(,)h Fp(jI)1655 4678 y Fk(2)1692 4666 y Fp(j)g Fs(and)f Fp(jI)1972 4678 y Fk(3)2010 4666 y Fp(j)p Fs(.)37 b(Using)27 b(that)h Fq(p)2548 4678 y Fk(4)2603 4666 y Fs(+)18 b(2)p Fq(l)24 b Fp(\025)f Fq(p)2907 4678 y Fk(3)2962 4666 y Fs(+)18 b(2,)27 b(w)n(e)g(see)g(that)h Fp(jI)3641 4678 y Fk(2)3678 4666 y Fp(j)g Fs(has)f(the)-260 4766 y(largest)f(b)r(ound)i(\(coming)g (from)f(the)h(extension)f(theorem\),)g(whic)n(h)h(giv)n(es)e(the)i(exp) r(ected)g(b)r(ound)g(for)f(\(53\).)779 b Fd(2)-260 4965 y Ft(Remark.)53 b Fs(The)28 b(case)e(of)h(\014xed)h(tori)e(is)h(a)g (bit)h(simpler,)f(b)r(ecause)g(in)g(this)h(case)e(it)i(is)f(enough)g (to)g(write)g(the)g(in)n(tegral)3493 4898 y Fj(R)3549 4919 y Fi(1)3533 4995 y Fk(0)3646 4965 y Fs(in)h(\(53\))f(as)-260 5065 y Fp(I)-209 5035 y Fi(0)-215 5085 y Fk(1)-159 5065 y Fs(+)18 b Fp(I)-25 5035 y Fi(0)-31 5085 y Fk(2)7 5065 y Fs(,)27 b(with)1195 5294 y Fp(I)1246 5260 y Fi(0)1240 5314 y Fk(1)1300 5294 y Fs(=)1388 5181 y Fj(Z)1471 5201 y Fl(t)1496 5209 y Fe(0)1528 5201 y Fk(\()p Fl(s)p Fk(\))1434 5369 y(0)1630 5227 y Fj(\002)1664 5294 y Fq(@)1708 5306 y Fl(')1756 5294 y Fq(H)1825 5306 y Fk(1)1876 5227 y Fj(\000)1914 5294 y Fq(Z)1977 5260 y Fk(+)2032 5227 y Fj(\001)2089 5294 y Fp(\000)18 b Fq(@)2216 5306 y Fl(')2263 5294 y Fq(H)2332 5306 y Fk(1)2384 5294 y Fs(\()p Fq(Z)2473 5306 y Fk(0)2510 5294 y Fs(\))2542 5227 y Fj(\003)2590 5294 y Fs(d)p Fq(t;)1195 5520 y Fp(I)1246 5486 y Fi(0)1240 5541 y Fk(2)1300 5520 y Fs(=)1388 5407 y Fj(Z)1471 5428 y Fi(1)1434 5596 y Fl(t)1459 5604 y Fe(0)1492 5596 y Fk(\()p Fl(s)p Fk(\))1593 5453 y Fj(\002)1627 5520 y Fq(@)1671 5532 y Fl(')1719 5520 y Fq(H)1788 5532 y Fk(1)1839 5453 y Fj(\000)1877 5520 y Fq(Z)1940 5486 y Fk(+)1995 5453 y Fj(\001)2052 5520 y Fp(\000)g Fq(@)2179 5532 y Fl(')2227 5520 y Fq(H)2296 5532 y Fk(1)2347 5520 y Fs(\()p Fq(Z)2436 5532 y Fk(0)2473 5520 y Fs(\))2505 5453 y Fj(\003)2554 5520 y Fs(d)p Fq(t:)-260 5740 y Fs(Ho)n(w)n(ev)n(er,)26 b(no)h(extra)g(impro)n(v)n(emen)n(t)f(in)i(the)g(b)r(ounds)g(is)f (obtained.)1825 5989 y(18)p eop %%Page: 19 19 19 18 bop -260 -82 a Fn(4.5)112 b(The)38 b(splitting)d(distance)i(in)g (the)g(\015o)m(w-b)s(o)m(x)i(co)s(ordinates)-260 171 y Fs(Next,)21 b(w)n(e)e(translate)f(the)i(parameterizations)d(of)i(the) g(whisk)n(ers)f(to)h(the)h(\015o)n(w-b)r(o)n(x)d(co)r(ordinates)h(\()p Fq(S;)c(E)5 b(;)14 b(';)g(I)7 b Fs(\))20 b(in)n(tro)r(duced)f(in)g (Section)h(3.)-260 270 y(Recall)27 b(that)h(our)f(Hamiltonian,)h(in)f (the)h(\015o)n(w-b)r(o)n(x)e(co)r(ordinates,)h(b)r(ecomes)1014 441 y Fq(K)1085 453 y Fk(0)1145 441 y Fs(=)22 b Fq(H)k Fp(\016)18 b Fs(\011)k(=)1578 420 y Fj(e)1562 441 y Fq(G)d Fp(\016)e Fs(\001)i Fp(\016)f Fs(\007)1918 406 y Fk(\(0\))2025 441 y Fp(\016)g Fs(\007)23 b(=)f Fp(h)7 b Fs(~)-48 b Fq(!)2345 453 y Fl(")2380 441 y Fq(;)14 b(I)7 b Fp(i)19 b Fs(+)f Fq(bE)5 b(;)-260 611 y Fs(on)27 b(the)h(domain)g Fp(B)349 623 y Fl(\024)388 631 y Fe(1)419 623 y Fl(;\033)477 631 y Fe(1)510 623 y Fl(;\021)564 631 y Fe(2)596 623 y Fl(;\032)650 631 y Fe(4)683 623 y Fl(;\014)741 631 y Fe(1)773 623 y Fl(\016)809 611 y Fs(.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)e(denoted)1422 792 y(\011)d(=)f(\000)d Fp(\016)f Fs(\010)g Fp(\016)g Fs(\001)h Fp(\016)f Fs(\007)2079 758 y Fk(\()o(0\))2186 792 y Fp(\016)g Fs(\007)1534 b(\(58\))-260 963 y(the)28 b(whole)f(transformation)f(from)i(the)g(original)e(co)r (ordinates)g(to)h(the)h(\015o)n(w-b)r(o)n(x)e(co)r(ordinates.)-135 1162 y(The)h(domain)h(of)f Fq(K)496 1174 y Fk(0)561 1162 y Fs(con)n(tains)f(a)h(piece)h(of)f(the)h Fo(lo)l(c)l(al)j(stable)f (whisker)f Fp(W)2203 1126 y Fk(+)2196 1187 y(lo)r(c)2284 1162 y Fs(,)f(whic)n(h)f(can)g(b)r(e)h(parameterized)e(as)1080 1332 y Fq(W)1170 1297 y Fk(+)1158 1354 y(0)1225 1332 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))24 b(=)e(\()p Fq(s;)14 b Fs(0)p Fq(;)g(\022)r(;)g Fs(0\))p Fq(;)180 b Fs(\()p Fq(s;)14 b(\022)r Fs(\))24 b Fp(2)f(P)2400 1344 y Fl(\024)2439 1352 y Fe(1)2471 1344 y Fl(;\033)2529 1352 y Fe(1)2562 1344 y Fl(;\032)2616 1352 y Fe(4)3845 1332 y Fs(\(59\))-260 1503 y(\(recall)30 b(the)i(notation)e(\(12\))h(for)f(the)h(domain)g(of) g(the)g(parameters)e(\()p Fq(s;)14 b(\022)r Fs(\)\).)48 b(The)31 b(inner)g(\015o)n(w)f(on)g(this)i(whisk)n(er)d(is)i(giv)n(en)f (b)n(y)43 b(_)-35 b Fq(s)28 b Fs(=)g Fq(b)p Fs(,)-244 1581 y(_)-260 1602 y Fq(\022)d Fs(=)30 b(~)-49 b Fq(!)-56 1614 y Fl(")-21 1602 y Fs(.)37 b(It)28 b(is)g(easy)e(to)i(establish)f (the)h(relation)f(b)r(et)n(w)n(een)g(this)h(parameterization)e(and)h (the)h(one)f(in)n(tro)r(duced)h(in)g(\(47\):)1320 1773 y(\011)1399 1706 y Fj(\000)1437 1773 y Fq(W)1527 1737 y Fk(+)1515 1795 y(0)1582 1773 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))1763 1706 y Fj(\001)1824 1773 y Fs(=)23 b Fq(Z)1975 1739 y Fk(+)2030 1773 y Fs(\()p Fq(s)18 b Fs(+)g Fq(s)2241 1785 y Fk(0)2279 1773 y Fq(;)c(\022)r Fs(\))p Fq(;)-260 1943 y Fs(This)28 b(comes)f(from)g(the)h(follo)n(wing)e(steps:)618 2150 y Fq(W)708 2115 y Fk(+)696 2172 y(0)763 2150 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))1009 2103 y Fk(\007)968 2150 y Fp(7\000)-14 b(!)23 b Fq(W)1215 2115 y Fk(+)1203 2172 y(0)1270 2150 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))1477 2103 y Fk(\007)1528 2078 y Fe(\()q(0\))1475 2150 y Fp(7\000)-15 b(!)23 b Fq(W)1721 2115 y Fk(+)1709 2172 y(0)1777 2150 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))2021 2103 y Fk(\001)1981 2150 y Fp(7\000)-14 b(!)23 b Fq(Y)2205 2115 y Fk(+)2186 2172 y(0)2260 2150 y Fs(\()p Fq(s)18 b Fs(+)h Fq(s)2472 2162 y Fk(0)2509 2150 y Fq(;)14 b(\022)r Fs(\))2685 2103 y Fk(\010)2642 2150 y Fp(7\000)-14 b(!)23 b Fq(Y)2866 2116 y Fk(+)2921 2150 y Fs(\()p Fq(s)18 b Fs(+)g Fq(s)3132 2162 y Fk(0)3170 2150 y Fq(;)c(\022)r Fs(\))831 2254 y Fk(\000)784 2301 y Fp(7\000)-14 b(!)23 b Fq(Z)1004 2266 y Fk(+)1059 2301 y Fs(\()p Fq(s)18 b Fs(+)h Fq(s)1271 2313 y Fk(0)1308 2301 y Fq(;)14 b(\022)r Fs(\))p Fq(;)2404 b Fs(\(60\))-260 2471 y(where)27 b(w)n(e)g(ha)n(v)n(e)g(used)g(that)h (\001)746 2404 y Fj(\000)784 2471 y Fq(W)874 2436 y Fk(+)862 2493 y(0)929 2471 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))1110 2404 y Fj(\001)1172 2471 y Fs(=)23 b(\(0)p Fq(;)14 b(v)1411 2483 y Fk(0)1448 2471 y Fq(e)1487 2441 y Fi(\000)p Fl(s)1574 2471 y Fq(;)g(\022)r(;)g Fs(0\))23 b(=)g Fq(Y)1940 2436 y Fk(+)1922 2493 y(0)1995 2471 y Fs(\()p Fq(s)c Fs(+)f Fq(s)2207 2483 y Fk(0)2244 2471 y Fq(;)c(\022)r Fs(\);)28 b(in)g(this)g(w)n(a)n(y)f(w)n(e)g(de\014ne)1615 2642 y Fq(s)1654 2654 y Fk(0)1714 2642 y Fs(=)c(ln\()p Fq(A=v)2047 2654 y Fk(0)2085 2642 y Fs(\))1728 b(\(61\))-260 2812 y(\(recall)30 b(that)g(in)h(\(45\))f(w)n(e)g(in)n(tro)r(duced)g(the)h (constan)n(t)e Fq(A)f Fs(=)f(4)1707 2744 y Fp(p)p 1776 2744 42 4 v 68 x Fs(2\))j(as)g(a)g(translation)f(b)r(et)n(w)n(een)h(b)r (oth)h(parameterizations,)e(in)i(order)-260 2912 y(to)f(ha)n(v)n(e)e Fq(W)127 2876 y Fk(+)115 2934 y(0)182 2912 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))30 b(de\014ned)g(around)f Fq(s)d Fs(=)g(0.)42 b(Recall)29 b(also)f(that)i(\007)1895 2882 y Fk(\(0\))2014 2912 y Fs(and)f(\007)g(lea)n(v)n(e)f(this)i(whisk)n(er) e(unc)n(hanged,)i(since)f(these)h(maps)-260 3012 y(are)d(id)18 b(+)g Fp(O)r Fs(\()p Fq(E)5 b(;)14 b(I)7 b Fs(\).)-135 3211 y(According)30 b(to)i(the)g(extension)f(theorem)g(\(Section)h (4.2\),)g(a)f(large)f(piece)h(of)h(the)g Fo(glob)l(al)j(unstable)e (whisker)g Fp(W)3420 3181 y Fi(\000)3507 3211 y Fs(remains)e(close)-260 3310 y(to)e(the)g(stable)f(one)g(for)g Fq(\026)h Fs(small)f(enough.)39 b(Then,)29 b(it)g(is)f(natural)g(to)h(de\014ne)f(the)h(follo)n(wing)f (parameterization)e(for)i(the)h(part)f(of)h(this)-260 3410 y(whisk)n(er)e(that)h(en)n(ters)e(in)i(the)g(domain)f(of)h(the)g (\015o)n(w-b)r(o)n(x)e(co)r(ordinates:)1256 3581 y Fq(W)1346 3546 y Fi(\000)1402 3581 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))24 b(:=)f(\011)1783 3546 y Fi(\000)p Fk(1)1885 3513 y Fj(\000)1923 3581 y Fq(Z)1986 3546 y Fi(\000)2042 3581 y Fs(\()p Fq(s)18 b Fs(+)h Fq(s)2254 3593 y Fk(0)2291 3581 y Fq(;)14 b(\022)r Fs(\))2401 3513 y Fj(\001)2453 3581 y Fq(:)1369 b Fs(\(62\))-260 3760 y(In)37 b(comp)r(onen)n(ts,)h(w) n(e)d(shall)h(write)g Fq(W)988 3730 y Fi(\000)1082 3760 y Fs(=)1184 3693 y Fj(\000)1222 3760 y Fq(W)1312 3725 y Fi(\000)1300 3785 y Fl(S)1368 3760 y Fq(;)14 b(W)1495 3725 y Fi(\000)1483 3785 y Fl(E)1551 3760 y Fq(;)g(W)1678 3730 y Fi(\000)1666 3781 y Fl(')1734 3760 y Fq(;)g(W)1861 3725 y Fi(\000)1849 3785 y Fl(I)1917 3693 y Fj(\001)1955 3760 y Fs(.)62 b(W)-7 b(e)37 b(are)e(going)g(to)i(establish)e(in)i (Prop)r(osition)e(9)g(that)i(this)-260 3860 y(parameterization)24 b(can)i(b)r(e)h(de\014ned)g(essen)n(tially)e(in)i(the)f(domain)g(giv)n (en)g(in)g(Theorem)g(6)g(and)g(Prop)r(osition)e(8)i(for)g(the)h (parameters)d Fq(s)-260 3960 y Fs(and)k Fq(\022)r Fs(,)g(with)h(a)e (reduction)h(of)g(only)g Fp(\030)23 b Fq(\016)31 b Fs(in)d(their)g (imaginary)e(parts.)38 b(Note)28 b(that)g(this)g(requires)f(to)h(tak)n (e)f(in)n(to)h(accoun)n(t)f(the)h(drastic)-260 4059 y(reduction)f(of)h (domain)f(for)g(the)h(\015o)n(w-b)r(o)n(x)e(co)r(ordinates,)g(in)i(the) g Fq(I)7 b Fs(-direction,)27 b(that)h(w)n(e)f(had)h(to)f(carry)f(out)i (in)g(Prop)r(osition)e(3.)-135 4258 y(Comparing)37 b(the)i Fq(I)7 b Fs(-comp)r(onen)n(ts)38 b(in)h(\(59\))g(and)f(\(62\),)k(w)n(e) c(can)h(consider)e(the)j(function)f Fq(W)2890 4223 y Fi(\000)2878 4283 y Fl(I)2946 4258 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))40 b(as)e(giving)g(the)h(splitting)-260 4358 y(distance.)i(Our)29 b(use)f(of)h(the)h(\015o)n(w-b)r(o)n(x)d(co)r (ordinates)h(implies)h(that)h(this)f(function)h(is)35 b(^)-48 b Fq(!)2526 4370 y Fl(")2561 4358 y Fs(-quasip)r(erio)r(dic)28 b(\(recall)g(that)36 b(^)-48 b Fq(!)3585 4370 y Fl(")3649 4358 y Fs(is)29 b(de\014ned)-260 4458 y(in)42 b(\(51\)\).)78 b(The)41 b(next)h(theorem)e(pro)n(vides)g(a)h(b)r(ound)h(for)f(this)g (function,)46 b(and)41 b(also)f(sa)n(ys)g(that)h(the)h(appro)n (ximation)d(giv)n(en)i(in)-260 4557 y(Prop)r(osition)26 b(8,)h(in)h(terms)f(of)g(the)h(Melnik)n(o)n(v)e(function)i Fq(M)9 b Fs(,)28 b(remains)e(true)i(after)f(c)n(hanging)f(from)h(the)h (original)d(co)r(ordinates)i(to)g(the)-260 4657 y(\015o)n(w-b)r(o)n(x) 32 b(co)r(ordinates.)52 b(In)34 b(fact,)h(since)e(the)h(translation)e (\(61\))h(in)g(the)h(v)-5 b(ariable)32 b Fq(s)h Fs(has)g(to)g(b)r(e)h (tak)n(en)f(in)n(to)g(accoun)n(t,)h(w)n(e)f(use)g(the)-260 4757 y(notation)27 b Fq(M)153 4769 y Fl(s)184 4777 y Fe(0)221 4757 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))23 b(=)g Fq(M)9 b Fs(\()p Fq(s)18 b Fs(+)g Fq(s)814 4769 y Fk(0)851 4757 y Fq(;)c(\022)r Fs(\).)-260 5027 y Ft(Prop)s(osition)30 b(9)42 b Fo(Under)31 b(c)l(onditions)i(\(26\))g(and)f(\(49\),)i(the)e (p)l(ar)l(ameterization)h Fq(W)2425 4997 y Fi(\000)2481 5027 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))33 b Fo(c)l(an)f(b)l(e)g (de\014ne)l(d)g(for)h Fs(\()p Fq(s;)14 b(\022)r Fs(\))27 b Fp(2)h(P)3720 5039 y Fl(\024)3759 5047 y Fe(2)3791 5039 y Fl(;\027)3844 5047 y Fe(2)3876 5039 y Fl(;\032)3930 5047 y Fe(5)3967 5027 y Fo(,)-260 5126 y(and)i(with)h(the)f(exp)l (onents)f Fq(p)642 5138 y Fl(i)699 5126 y Fo(de\014ne)l(d)h(in)g (\(14{19\),)i(one)e(has:)-200 5432 y Fs(\(a\))-52 5362 y Fj(\014)-52 5411 y(\014)-25 5432 y Fq(W)65 5398 y Fi(\000)140 5432 y Fp(\000)18 b Fq(W)313 5397 y Fk(+)301 5454 y(0)368 5362 y Fj(\014)368 5411 y(\014)396 5465 y Fl(\024)435 5473 y Fe(2)467 5465 y Fl(;\027)520 5473 y Fe(2)552 5465 y Fl(;\032)606 5473 y Fe(5)666 5432 y Fp(\026)754 5315 y Fj(\022)855 5376 y Fq(\026)p 825 5413 111 4 v 825 5489 a(\016)865 5465 y Fl(p)899 5473 y Fe(5)964 5432 y Fs(+)1141 5376 y Fq(\026)p 1057 5413 219 4 v 1057 5489 a(\016)1097 5465 y Fl(p)1131 5473 y Fe(2)1168 5430 y Fp(p)p 1237 5430 39 4 v 59 x Fq(")1286 5432 y(;)1363 5376 y(\026)p 1332 5413 111 4 v 1332 5489 a(\016)1372 5465 y Fl(p)1406 5473 y Fe(4)1472 5432 y Fs(+)1649 5376 y Fq(\026)p 1565 5413 219 4 v 1565 5489 a(\016)1605 5465 y Fl(p)1639 5473 y Fe(2)1676 5430 y Fp(p)p 1745 5430 39 4 v 59 x Fq(")1793 5315 y Fj(\023)1854 5432 y Fo(.)-204 5705 y Fs(\(b\))-52 5634 y Fj(\014)-52 5684 y(\014)-25 5705 y Fq(W)65 5669 y Fi(\000)53 5729 y Fl(I)140 5705 y Fp(\000)g Fq(\026M)354 5717 y Fl(s)385 5725 y Fe(0)421 5634 y Fj(\014)421 5684 y(\014)449 5738 y Fl(\024)488 5746 y Fe(2)520 5738 y Fl(;\027)573 5746 y Fe(2)606 5738 y Fl(;\032)660 5746 y Fe(5)719 5705 y Fp(\026)829 5648 y Fq(\026)879 5618 y Fk(2)p 817 5685 111 4 v 817 5762 a Fq(\016)857 5738 y Fl(p)891 5746 y Fe(6)956 5705 y Fs(+)1115 5648 y Fq(\026)1165 5618 y Fk(2)p 1049 5685 219 4 v 1049 5762 a Fq(\016)1089 5738 y Fl(p)1123 5746 y Fe(7)1160 5702 y Fp(p)p 1229 5702 39 4 v 60 x Fq(")1278 5705 y Fo(.)1825 5989 y Fs(19)p eop %%Page: 20 20 20 19 bop -195 -82 a Fs(\(c\))42 b Fq(W)38 -112 y Fi(\000)94 -82 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))19 b Fp(\000)f Fq(W)467 -118 y Fk(+)455 -60 y(0)522 -82 y Fs(\()p Fq(s;)c(\022)r Fs(\))24 b(=)814 -149 y Fj(\000)852 -82 y Fq(W)942 -118 y Fi(\000)930 -58 y Fl(S)1017 -82 y Fp(\000)18 b Fq(s;)c(W)1266 -118 y Fi(\000)1254 -58 y Fl(E)1322 -82 y Fq(;)g(W)1449 -112 y Fi(\000)1437 -62 y Fl(')1523 -82 y Fp(\000)k Fq(\022)r(;)c(W) 1774 -118 y Fi(\000)1762 -58 y Fl(I)1830 -149 y Fj(\001)1898 -82 y Fo(is)36 b Fs(^)-48 b Fq(!)2039 -70 y Fl(")2074 -82 y Fo(-quasip)l(erio)l(dic.)-260 211 y(Pr)l(o)l(of.)50 b Fs(First,)25 b(w)n(e)e(p)r(oin)n(t)i(out)f(that)g(the)g(\015o)n(w-b)r (o)n(x)f(co)r(ordinates)g(can)g(b)r(e)i(de\014ned)f(if)h(conditions)e (\(26\))h(and)g(\(43\))g(are)f(ful\014lled.)36 b(Notice)-260 311 y(that)28 b(condition)f(\(43\),)h(needed)f(in)h(Theorem)f(4,)g(is)h (included)g(in)g(condition)f(\(49\),)g(required)g(here.)-135 510 y(T)-7 b(o)25 b(pro)n(v)n(e)g(part)g(\(a\),)i(w)n(e)f(ha)n(v)n(e)f (to)h(consider)f(the)h(e\013ect)h(of)f(the)g(successiv)n(e)f (transformations)f(\(58\))i(on)g(the)g(b)r(ounds)g(pro)n(vided)f(b)n(y) -260 610 y(Theorem)32 b(6)g(and)g(Prop)r(osition)f(8.)51 b(In)33 b(order)e(to)i(write)f(the)h(successiv)n(e)e(b)r(ounds)i(in)g (a)f(simple)g(w)n(a)n(y)-7 b(,)33 b(w)n(e)f(in)n(tro)r(duce)g(the)h (follo)n(wing)-260 709 y(notations)27 b(for)g(the)h(in)n(termediate)f (steps)h(in)g(\(62\),)f(to)g(b)r(e)h(compared)f(with)h(\(60\):)769 937 y Fq(Z)832 903 y Fi(\000)888 937 y Fs(\()p Fq(s)18 b Fs(+)g Fq(s)1099 949 y Fk(0)1137 937 y Fq(;)c(\022)r Fs(\))1278 890 y Fk(\000)1319 865 y Fh(\000)p Fe(1)1270 937 y Fp(7\000)-14 b(!)23 b Fq(Y)1494 901 y Fi(\000)1475 965 y Fk(\(1\))1564 937 y Fs(\()p Fq(s)c Fs(+)f Fq(s)1776 949 y Fk(0)1813 937 y Fq(;)c(\022)r Fs(\))1951 890 y Fk(\010)1998 865 y Fh(\000)p Fe(1)1947 937 y Fp(7\000)-15 b(!)23 b Fq(Y)2170 901 y Fi(\000)2151 965 y Fk(\(2\))2241 937 y Fs(\()p Fq(s)18 b Fs(+)g Fq(s)2452 949 y Fk(0)2490 937 y Fq(;)c(\022)r Fs(\))2624 890 y Fk(\001)2679 865 y Fh(\000)p Fe(1)2623 937 y Fp(7\000)-14 b(!)23 b Fq(W)2870 901 y Fi(\000)2858 965 y Fk(\(1\))2948 937 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))935 1091 y(\()967 1085 y Fk(\007)1018 1060 y Fe(\()q(0\))1096 1091 y Fs(\))1128 1049 y Fh(\000)p Fe(1)1003 1151 y Fp(7\000)-14 b(!)92 b Fq(W)1319 1116 y Fi(\000)1307 1180 y Fk(\(2\))1396 1151 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))1603 1104 y Fk(\007)1654 1079 y Fh(\000)p Fe(1)1601 1151 y Fp(7\000)-14 b(!)23 b Fq(W)1848 1117 y Fi(\000)1904 1151 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))p Fq(:)-135 1450 y Fs(W)-7 b(e)28 b(b)r(egin)f(with)h(the)g(follo)n(wing) f(b)r(ound,)h(giv)n(en)f(b)n(y)g(Theorem)g(6:)911 1570 y Fj(\014)911 1619 y(\014)939 1640 y Fq(Z)1002 1606 y Fi(\000)1058 1640 y Fs(\()p Fq(s)18 b Fs(+)g Fq(s)1269 1652 y Fk(0)1306 1640 y Fq(;)c(\022)r Fs(\))19 b Fp(\000)f Fq(Z)1581 1606 y Fk(+)1636 1640 y Fs(\()p Fq(s)h Fs(+)f Fq(s)1848 1652 y Fk(0)1885 1640 y Fq(;)c(\022)r Fs(\))1995 1570 y Fj(\014)1995 1619 y(\014)2046 1640 y Fp(\026)2134 1570 y Fj(\014)2134 1619 y(\014)2162 1640 y Fq(Z)2225 1606 y Fi(\000)2299 1640 y Fp(\000)k Fq(Z)2439 1652 y Fk(0)2476 1570 y Fj(\014)2476 1619 y(\014)2522 1640 y Fs(+)2605 1570 y Fj(\014)2605 1619 y(\014)2632 1640 y Fq(Z)2695 1606 y Fk(+)2769 1640 y Fp(\000)g Fq(Z)2909 1652 y Fk(0)2945 1570 y Fj(\014)2945 1619 y(\014)1077 1823 y Fp(\026)1165 1706 y Fj(\022)1266 1767 y Fq(\026)p 1236 1804 111 4 v 1236 1880 a(\016)1276 1856 y Fl(p)1310 1864 y Fe(5)1357 1823 y Fq(;)1434 1767 y(\026)p 1404 1804 V 1404 1880 a(\016)1444 1856 y Fl(p)1478 1864 y Fe(4)1543 1823 y Fs(+)1720 1767 y Fq(\026)p 1636 1804 219 4 v 1636 1880 a(\016)1676 1856 y Fl(p)1710 1864 y Fe(2)1747 1820 y Fp(p)p 1816 1820 39 4 v 60 x Fq(")1865 1706 y Fj(\023)1940 1823 y Fq(;)1882 b Fs(\(63\))1243 1997 y Fp(j)p Fs(Re\()p Fq(s)19 b Fs(+)f Fq(s)1576 2009 y Fk(0)1613 1997 y Fs(\))p Fp(j)24 b(\024)e Fq(q)1816 2009 y Fi(\003)1855 1997 y Fq(;)41 b Fp(j)p Fs(Im)14 b Fq(s)p Fp(j)23 b(\024)g Fq(\027)2269 2009 y Fk(1)2306 1997 y Fq(;)42 b Fp(j)p Fs(Im)14 b Fq(\022)r Fp(j)23 b(\024)g Fq(\032)2725 2009 y Fk(3)2762 1997 y Fq(:)-260 2187 y Fs(Let)33 b(us)f(study)h(the)g(e\013ect)g(of)g(the)g(successiv)n (e)e(transformations)f(on)j(this)f(b)r(ound.)53 b(The)32 b(k)n(ey)g(p)r(oin)n(ts,)i(in)f(order)e(to)h(sho)n(w)g(that)h(the)-260 2287 y(\014nal)c(b)r(ound)f(do)r(es)h(not)f(get)h(essen)n(tially)e(w)n (orse)g(than)i(\(63\),)f(are)g(that)g(\000)2053 2257 y Fi(\000)p Fk(1)2171 2287 y Fs(and)g(\001)2402 2257 y Fi(\000)p Fk(1)2520 2287 y Fs(do)h(not)f(touc)n(h)g(the)h Fq(I)7 b Fs(-comp)r(onen)n(t,)29 b(and)f(that)-260 2407 y(\010)-200 2376 y Fi(\000)p Fk(1)-83 2407 y Fs(and)f(\007)143 2376 y Fi(\000)p Fk(1)260 2407 y Fs(are)f(close)h(to)h(the)g(iden)n (tit)n(y)-7 b(.)37 b(On)27 b(the)h(other)f(hand,)h(the)g (transformation)2612 2339 y Fj(\000)2650 2407 y Fs(\007)2715 2376 y Fk(\()o(0\))2803 2339 y Fj(\001)2841 2353 y Fi(\000)p Fk(1)2958 2407 y Fs(will)g(easily)f(b)r(e)h(con)n(trolled.)-135 2606 y(First,)46 b(recall)c(the)h(form)f(of)h(the)g(c)n(hange)e(to)i (the)g(h)n(yp)r(erb)r(olic)f(co)r(ordinates:)66 b(\000\()p Fq(u;)14 b(v)s(;)g(';)g(I)7 b Fs(\))48 b(=)g(\(\000)3132 2618 y Fk(0)3169 2606 y Fs(\()p Fq(u;)14 b(v)s Fs(\))p Fq(;)g(';)g(I)7 b Fs(\).)83 b(W)-7 b(e)43 b(ha)n(v)n(e)-260 2635 y Fj(\014)-260 2685 y(\014)-232 2705 y Fs(D)-155 2638 y Fj(\000)-117 2705 y Fs(\000)-65 2670 y Fi(\000)p Fk(1)-65 2728 y(0)24 2638 y Fj(\001)62 2635 y(\014)62 2685 y(\014)113 2705 y Fp(\026)23 b Fs(1)k(on)g(its)h(domain,)f(and)h (then)g(from)f(\(63\))g(w)n(e)g(directly)h(obtain)845 2858 y Fj(\014)845 2908 y(\014)845 2958 y(\014)872 2954 y Fq(Y)939 2918 y Fi(\000)920 2982 y Fk(\(1\))1010 2954 y Fs(\()p Fq(s)18 b Fs(+)g Fq(s)1221 2966 y Fk(0)1259 2954 y Fq(;)c(\022)r Fs(\))k Fp(\000)g Fq(Y)1537 2919 y Fk(+)1592 2954 y Fs(\()p Fq(s)h Fs(+)f Fq(s)1804 2966 y Fk(0)1841 2954 y Fq(;)c(\022)r Fs(\))1951 2858 y Fj(\014)1951 2908 y(\014)1951 2958 y(\014)2002 2954 y Fp(\026)2090 2837 y Fj(\022)2191 2898 y Fq(\026)p 2161 2935 111 4 v 2161 3011 a(\016)2201 2987 y Fl(p)2235 2995 y Fe(5)2282 2954 y Fq(;)2359 2898 y(\026)p 2329 2935 V 2329 3011 a(\016)2369 2987 y Fl(p)2403 2995 y Fe(4)2468 2954 y Fs(+)2645 2898 y Fq(\026)p 2561 2935 219 4 v 2561 3011 a(\016)2601 2987 y Fl(p)2635 2995 y Fe(2)2672 2951 y Fp(p)p 2741 2951 39 4 v 60 x Fq(")2790 2837 y Fj(\023)2864 2954 y Fq(:)958 b Fs(\(64\))-260 3189 y(Next,)23 b(w)n(e)e(consider)f (the)h(transformation)f(to)h(the)g(normal)f(form)h(co)r(ordinates,)g (pro)n(vided)f(b)n(y)h(Theorem)g(1.)34 b(F)-7 b(or)20 b(this)i(transformation,)-260 3289 y(w)n(e)31 b(easily)f(see)g(from)g (\(29\))h(that)g Fp(j)p Fs(\010)18 b Fp(\000)h Fs(id)p Fp(j)28 b(\026)g Fs(\(1)p Fq(;)14 b(\016)s Fs(\))31 b(and)g(w)n(e)f(ha) n(v)n(e)g(\010)1953 3258 y Fi(\000)p Fk(1)2070 3289 y Fs(:)f Fp(S)2172 3301 y Fl(r)2203 3309 y Fe(2)2235 3301 y Fl(;\015)2290 3309 y Fe(2)2322 3301 y Fl(;\032)2376 3309 y Fe(3)2441 3289 y Fp(\000)-14 b(!)28 b(S)2653 3301 y Fl(r)2684 3309 y Fe(1)2717 3301 y Fl(;\015)2772 3309 y Fe(1)2804 3301 y Fl(;\032)2858 3309 y Fe(2)2895 3289 y Fs(.)46 b(W)-7 b(riting)31 b(\010)3332 3258 y Fi(\000)p Fk(1)3450 3289 y Fs(=)d(id)20 b(+)h Fq(\026)3777 3268 y Fs(^)3768 3289 y(\010)30 b(and)-260 3388 y(applying)d(no)n(w)g (\(29\))g(to)529 3318 y Fj(\014)529 3367 y(\014)557 3388 y Fs(\010)617 3358 y Fi(\000)p Fk(1)724 3388 y Fp(\000)18 b Fs(id)876 3318 y Fj(\014)876 3367 y(\014)904 3388 y Fs(,)28 b(w)n(e)f(obtain)1199 3541 y Fj(\014)1199 3591 y(\014)1199 3641 y(\014)1227 3636 y Fq(\026)p Fs(D)1349 3615 y(^)1340 3636 y(\010)1400 3541 y Fj(\014)1400 3591 y(\014)1400 3641 y(\014)1428 3695 y Fl(r)1459 3703 y Fe(3)1491 3695 y Fl(;\015)1546 3703 y Fe(3)1578 3695 y Fl(;\032)1632 3703 y Fe(4)1692 3636 y Fp(\026)1820 3580 y Fq(\026)p 1790 3617 111 4 v 1790 3693 a(\016)1830 3669 y Fl(p)1864 3677 y Fe(3)1924 3519 y Fj(\022)2068 3586 y Fs(1)142 b(1)p Fq(=\016)2027 3685 y Fs(1)p Fq(=\016)85 b Fs(1)p Fq(=\016)2357 3655 y Fk(2)2435 3519 y Fj(\023)2510 3636 y Fq(;)1312 b Fs(\(65\))-260 3910 y(where)25 b(w)n(e)h(use)f(the)i (notation)e(of)h(\(56\).)36 b(Then,)26 b(in)g(the)g(same)f(domain)h(w)n (e)f(ha)n(v)n(e)2296 3839 y Fj(\014)2296 3889 y(\014)2323 3910 y Fs(D)2400 3842 y Fj(\000)2439 3910 y Fs(\010)2499 3879 y Fi(\000)p Fk(1)2588 3842 y Fj(\001)2626 3839 y(\014)2626 3889 y(\014)2677 3910 y Fp(\024)d Fs(Id)15 b(+)2935 3814 y Fj(\014)2935 3864 y(\014)2935 3914 y(\014)2962 3910 y Fq(\026)p Fs(D)3084 3889 y(^)3075 3910 y(\010)3136 3814 y Fj(\014)3136 3864 y(\014)3136 3914 y(\014)3163 3910 y Fs(.)37 b(Applying)26 b(this)g(b)r(ound,)-260 4029 y(w)n(e)h(obtain)h(again)845 4090 y Fj(\014)845 4140 y(\014)845 4190 y(\014)872 4185 y Fq(Y)939 4150 y Fi(\000)920 4214 y Fk(\(2\))1010 4185 y Fs(\()p Fq(s)18 b Fs(+)g Fq(s)1221 4197 y Fk(0)1259 4185 y Fq(;)c(\022)r Fs(\))k Fp(\000)g Fq(Y)1537 4150 y Fk(+)1518 4208 y(0)1592 4185 y Fs(\()p Fq(s)h Fs(+)f Fq(s)1804 4197 y Fk(0)1841 4185 y Fq(;)c(\022)r Fs(\))1951 4090 y Fj(\014)1951 4140 y(\014)1951 4190 y(\014)2002 4185 y Fp(\026)2090 4068 y Fj(\022)2191 4129 y Fq(\026)p 2161 4166 V 2161 4242 a(\016)2201 4218 y Fl(p)2235 4226 y Fe(5)2282 4185 y Fq(;)2359 4129 y(\026)p 2329 4166 V 2329 4242 a(\016)2369 4218 y Fl(p)2403 4226 y Fe(4)2468 4185 y Fs(+)2645 4129 y Fq(\026)p 2561 4166 219 4 v 2561 4243 a(\016)2601 4219 y Fl(p)2635 4227 y Fe(2)2672 4183 y Fp(p)p 2741 4183 39 4 v 60 x Fq(")2790 4068 y Fj(\023)2864 4185 y Fq(:)-260 4401 y Fs(The)22 b(c)n(hange)f(\001)243 4371 y Fi(\000)p Fk(1)355 4401 y Fs(to)h(the)h(unp)r(erturb)r(ed)f(\015o)n(w-b)r(o)n(x)f (co)r(ordinates)g(w)n(orks)f(lik)n(e)i(\000)2255 4371 y Fi(\000)p Fk(1)2344 4401 y Fs(,)i(and)e(then)g(the)h(b)r(ound)g (remains)e(also)g(unc)n(hanged:)982 4541 y Fj(\014)982 4591 y(\014)982 4641 y(\014)1009 4636 y Fq(W)1099 4601 y Fi(\000)1087 4665 y Fk(\(1\))1191 4636 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))19 b Fp(\000)f Fq(W)1564 4601 y Fk(+)1552 4659 y(0)1633 4636 y Fs(\()p Fq(s;)c(\022)r Fs(\))1815 4541 y Fj(\014)1815 4591 y(\014)1815 4641 y(\014)1865 4636 y Fp(\026)1953 4519 y Fj(\022)2054 4580 y Fq(\026)p 2024 4617 111 4 v 2024 4693 a(\016)2064 4669 y Fl(p)2098 4677 y Fe(5)2145 4636 y Fq(;)2222 4580 y(\026)p 2192 4617 V 2192 4693 a(\016)2232 4669 y Fl(p)2266 4677 y Fe(4)2331 4636 y Fs(+)2508 4580 y Fq(\026)p 2424 4617 219 4 v 2424 4694 a(\016)2464 4670 y Fl(p)2498 4678 y Fe(2)2535 4634 y Fp(p)p 2604 4634 39 4 v 60 x Fq(")2653 4519 y Fj(\023)2727 4636 y Fq(:)-260 4905 y Fs(Next)28 b(w)n(e)g(consider)f(the)h(transformation)1099 4838 y Fj(\000)1137 4905 y Fs(\007)1202 4875 y Fk(\(0\))1291 4838 y Fj(\001)1329 4852 y Fi(\000)p Fk(1)1418 4905 y Fs(,)g(whic)n(h)g(can)f(explicitly)h(b)r(e)g(written)g(from)g(\(38\),)g (and)f(the)h Fp(O)3389 4917 y Fk(2)3427 4905 y Fs(\()p Fq(I)7 b Fs(\)-term)28 b(in)g(this)-260 5004 y(transformation)e(pro)n (vides)g(a)h(new)h(term)g(in)g(our)e(b)r(ound.)38 b(Indeed,)28 b(denoting)f(\()p Fp(\001)p Fs(\))2327 5029 y Fl(S)2376 5004 y Fs(,)g(\()q Fp(\001)p Fs(\))2514 5029 y Fl(I)2580 5004 y Fs(the)h Fq(S;)14 b(I)7 b Fs(-comp)r(onen)n(ts,)26 b(w)n(e)i(ha)n(v)n(e)526 5248 y Fq(W)616 5213 y Fi(\000)604 5277 y Fk(\(2\))712 5248 y Fp(\000)18 b Fq(W)885 5213 y Fi(\000)873 5277 y Fk(\(1\))986 5248 y Fs(=)1073 5131 y Fj(\022)1135 5248 y Fs(0)p Fq(;)1241 5192 y Fs(1)p 1223 5229 78 4 v 1223 5305 a(2)p Fq(b)1324 5156 y Fj(D)1375 5248 y Fs(\003)1447 5156 y Fj(\020)1496 5248 y Fq(W)1586 5213 y Fi(\000)1574 5277 y Fk(\(1\))1663 5156 y Fj(\021)1713 5307 y Fl(I)1765 5248 y Fq(;)1802 5156 y Fj(\020)1851 5248 y Fq(W)1941 5213 y Fi(\000)1929 5277 y Fk(\(1\))2019 5156 y Fj(\021)2068 5307 y Fl(I)2106 5156 y Fj(E)2171 5248 y Fq(;)c Fp(\000)2283 5192 y Fs(1)p 2283 5229 42 4 v 2286 5305 a Fq(b)2348 5156 y Fj(\020)2397 5248 y Fq(W)2487 5213 y Fi(\000)2475 5277 y Fk(\(1\))2565 5156 y Fj(\021)2614 5307 y Fl(S)2676 5248 y Fs(\003)2748 5156 y Fj(\020)2797 5248 y Fq(W)2887 5213 y Fi(\000)2875 5277 y Fk(\(1\))2965 5156 y Fj(\021)3015 5307 y Fl(I)3067 5248 y Fq(;)g Fs(0)3146 5131 y Fj(\023)-260 5484 y Fs(and)28 b(w)n(e)f(deduce)h(the)g(b)r(ound)812 5607 y Fj(\014)812 5657 y(\014)812 5707 y(\014)839 5703 y Fq(W)929 5667 y Fi(\000)917 5731 y Fk(\(2\))1021 5703 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))19 b Fp(\000)f Fq(W)1394 5667 y Fk(+)1382 5725 y(0)1463 5703 y Fs(\()p Fq(s;)c(\022)r Fs(\))1644 5607 y Fj(\014)1644 5657 y(\014)1644 5707 y(\014)1695 5703 y Fp(\026)1783 5586 y Fj(\022)1884 5647 y Fq(\026)p 1854 5684 111 4 v 1854 5760 a(\016)1894 5736 y Fl(p)1928 5744 y Fe(5)1993 5703 y Fs(+)2170 5647 y Fq(\026)p 2086 5684 219 4 v 2086 5760 a(\016)2126 5736 y Fl(p)2160 5744 y Fe(2)2197 5700 y Fp(p)p 2266 5700 39 4 v 60 x Fq(")2315 5703 y(;)2392 5647 y(\026)p 2362 5684 111 4 v 2362 5760 a(\016)2402 5736 y Fl(p)2436 5744 y Fe(4)2501 5703 y Fs(+)2678 5647 y Fq(\026)p 2594 5684 219 4 v 2594 5760 a(\016)2634 5736 y Fl(p)2668 5744 y Fe(2)2705 5700 y Fp(p)p 2774 5700 39 4 v 60 x Fq(")2823 5586 y Fj(\023)2898 5703 y Fq(:)924 b Fs(\(66\))1825 5989 y(20)p eop %%Page: 21 21 21 20 bop -260 -82 a Fs(Finally)-7 b(,)33 b(for)f(the)g(transformation) f(\007)959 -112 y Fi(\000)p Fk(1)1079 -82 y Fs(pro)n(vided)g(b)n(y)h (Theorem)f(4,)i(w)n(e)f(pro)r(ceed)f(as)h(with)g(\010)2796 -112 y Fi(\000)p Fk(1)2885 -82 y Fs(.)51 b(So)31 b(w)n(e)h(write)g (\007)3487 -112 y Fi(\000)p Fk(1)3606 -82 y Fs(=)e(id)22 b(+)f Fq(\026)3940 -103 y Fs(^)3928 -82 y(\007)-260 17 y(and,)28 b(applying)f(\(41\))g(and)g(\(42\))h(to)877 -53 y Fj(\014)877 -3 y(\014)905 17 y Fs(\007)970 -13 y Fi(\000)p Fk(1)1077 17 y Fp(\000)18 b Fs(id)1229 -53 y Fj(\014)1229 -3 y(\014)1257 17 y Fs(,)28 b(w)n(e)f(obtain)1276 180 y Fj(\014)1276 229 y(\014)1276 279 y(\014)1303 275 y Fq(\026)p Fs(D)1428 254 y(^)1416 275 y(\007)1481 180 y Fj(\014)1481 229 y(\014)1481 279 y(\014)1532 275 y Fp(\026)1702 219 y Fq(\026)p 1630 256 195 4 v 1630 332 a(\016)1670 308 y Fl(p)1704 316 y Fe(3)1736 308 y Fk(+1)1848 158 y Fj(\022)1992 224 y Fs(1)142 b(1)p Fq(=\016)1951 324 y Fs(1)p Fq(=\016)85 b Fs(1)p Fq(=\016)2281 294 y Fk(2)2359 158 y Fj(\023)2434 275 y Fq(;)1388 b Fs(\(67\))-260 558 y(and)28 b(hence)132 487 y Fj(\014)132 537 y(\014)160 558 y Fs(D)237 490 y Fj(\000)275 558 y Fs(\007)340 528 y Fi(\000)p Fk(1)429 490 y Fj(\001)467 487 y(\014)467 537 y(\014)518 558 y Fp(\024)22 b Fs(Id)d(+)783 462 y Fj(\014)783 512 y(\014)783 562 y(\014)811 558 y Fq(\026)p Fs(D)936 537 y(^)924 558 y(\007)989 462 y Fj(\014)989 512 y(\014)989 562 y(\014)1016 558 y Fs(.)37 b(Applying)28 b(this)g(to)f(\(66\),)h(w)n(e)f(obtain)g(the)h(b)r(ound)g(of)g(part)f (\(a\).)-135 786 y(W)-7 b(e)35 b(also)f(ha)n(v)n(e)g(to)h(establish)g (the)g(condition)g(on)g Fq(\026)p Fs(,)i(for)d(whic)n(h)h Fq(W)2071 756 y Fi(\000)2127 786 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))36 b(can)f(actually)f(b)r(e)i(de\014ned)f(in)h Fp(P)3404 798 y Fl(\024)3443 806 y Fe(2)3475 798 y Fl(;\027)3528 806 y Fe(2)3560 798 y Fl(;\032)3614 806 y Fe(5)3651 786 y Fs(,)h(without)-260 885 y(lea)n(ving)25 b(the)i(domain)e(of)h Fq(K)619 897 y Fk(0)656 885 y Fs(.)37 b(The)26 b(only)g(imp)r(ortan)n (t)f(restriction)g(comes)h(from)g(Prop)r(osition)e(3,)i(where)g(the)g (domain)g(in)g(the)h(actions)-260 1005 y Fq(I)40 b Fs(shrinks)31 b(to)i(a)f(width)h Fp(\030)e Fq(\016)s Fs(.)52 b(In)32 b(order)g(to)g(k)n(eep)g Fq(W)1458 970 y Fi(\000)1446 1034 y Fk(\(2\))1536 1005 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))32 b(=)1844 938 y Fj(\000)1882 1005 y Fs(\007)1947 975 y Fk(\(0\))2036 938 y Fj(\001)2074 952 y Fi(\000)p Fk(1)2177 913 y Fj(\020)2227 1005 y Fq(W)2317 970 y Fi(\000)2305 1034 y Fk(\(1\))2394 1005 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))2575 913 y Fj(\021)2658 1005 y Fs(in)33 b(the)g(domain,)g (according)e(to)h(\(66\))h(w)n(e)-260 1125 y(clearly)27 b(need)1586 1211 y Fq(\026)p 1556 1248 111 4 v 1556 1324 a(\016)1596 1300 y Fl(p)1630 1308 y Fe(4)1695 1267 y Fs(+)1873 1211 y Fq(\026)p 1788 1248 219 4 v 1788 1324 a(\016)1828 1300 y Fl(p)1862 1308 y Fe(2)1899 1264 y Fp(p)p 1968 1264 39 4 v 60 x Fq(")2040 1267 y Fp(\026)c Fq(\016)o(;)-260 1490 y Fs(and)28 b(this)f(condition)h(is)f(included)h (in)g(\(49\).)-135 1689 y(T)-7 b(o)31 b(obtain)g(part)f(\(b\),)k(w)n(e) d(ha)n(v)n(e)f(to)h(relate)f(the)i(function)g Fq(W)1829 1654 y Fi(\000)1817 1714 y Fl(I)1916 1689 y Fs(with)g(the)g (di\013erence)f Fq(Z)2694 1654 y Fi(\000)2688 1714 y Fl(I)2771 1689 y Fp(\000)20 b Fq(Z)2919 1654 y Fk(+)2913 1714 y Fl(I)3005 1689 y Fs(that)32 b(has)f(b)r(een)h(estimated)f(in) -260 1789 y(Prop)r(osition)f(8.)49 b(This)32 b(is)f(quite)h(simple)g(b) r(ecause)g(the)g(only)f(t)n(w)n(o)g(transformations)f(that)i(touc)n(h)f (the)i Fq(I)7 b Fs(-comp)r(onen)n(t)31 b(are)f(\010)3737 1759 y Fi(\000)p Fk(1)3858 1789 y Fs(and)-260 1888 y(\007)-195 1858 y Fi(\000)p Fk(1)-106 1888 y Fs(.)63 b(W)-7 b(e)37 b(write)354 1821 y Fj(\000)392 1888 y Fs(\010)452 1858 y Fi(\000)p Fk(1)541 1821 y Fj(\001)579 1922 y Fl(I)655 1888 y Fs(=)g Fq(I)32 b Fs(+)24 b Fq(\026)973 1867 y Fs(^)964 1888 y(\010)1024 1900 y Fl(I)1098 1888 y Fs(and)1268 1821 y Fj(\000)1307 1888 y Fs(\007)1372 1858 y Fi(\000)p Fk(1)1460 1821 y Fj(\001)1498 1922 y Fl(I)1574 1888 y Fs(=)38 b Fq(I)31 b Fs(+)24 b Fq(\026)1895 1867 y Fs(^)1883 1888 y(\007)1948 1900 y Fl(I)2022 1888 y Fs(\(the)37 b(subscript)g(\()p Fp(\001)p Fs(\))2659 1900 y Fl(I)2734 1888 y Fs(alw)n(a)n(ys)d(denotes)j(the)f Fq(I)7 b Fs(-comp)r(onen)n (t\).)-260 2001 y(Recalling)27 b(that)281 1934 y Fj(\000)319 2001 y Fq(W)409 1966 y Fk(+)397 2024 y(0)464 1934 y Fj(\001)502 2035 y Fl(I)563 2001 y Fs(=)c(0)k(and)882 1934 y Fj(\000)920 2001 y Fq(Y)986 1966 y Fk(+)968 2024 y(0)1041 1934 y Fj(\001)1079 2035 y Fl(I)1141 2001 y Fs(=)22 b(0,)27 b(w)n(e)h(ha)n(v)n(e)615 2238 y Fq(W)705 2203 y Fi(\000)693 2263 y Fl(I)761 2238 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))84 b(=)1173 2146 y Fj(\020)1223 2238 y Fq(W)1313 2203 y Fi(\000)1301 2267 y Fk(\(2\))1391 2146 y Fj(\021)1440 2296 y Fl(I)1492 2238 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))19 b(+)f Fq(\026)1839 2146 y Fj(h)1890 2217 y Fs(^)1878 2238 y(\007)1943 2250 y Fl(I)1995 2146 y Fj(\020)2044 2238 y Fq(W)2134 2203 y Fi(\000)2122 2267 y Fk(\(2\))2212 2238 y Fs(\()p Fq(s;)c(\022)r Fs(\))2393 2146 y Fj(\021)2462 2238 y Fp(\000)2556 2217 y Fs(^)2545 2238 y(\007)2610 2250 y Fl(I)2661 2171 y Fj(\000)2699 2238 y Fq(W)2789 2203 y Fk(+)2777 2260 y(0)2844 2238 y Fs(\()p Fq(s;)g(\022)r Fs(\))3025 2171 y Fj(\001)3064 2146 y(i)1026 2425 y Fs(=)1173 2333 y Fj(\020)1223 2425 y Fq(W)1313 2390 y Fi(\000)1301 2453 y Fk(\(1\))1391 2333 y Fj(\021)1440 2483 y Fl(I)1492 2425 y Fs(\()p Fq(s;)g(\022)r Fs(\))19 b(+)f Fq(\026)1839 2333 y Fj(h)1890 2404 y Fs(^)1878 2425 y(\007)1943 2437 y Fl(I)1995 2333 y Fj(\020)2044 2425 y Fq(W)2134 2390 y Fi(\000)2122 2453 y Fk(\(2\))2212 2425 y Fs(\()p Fq(s;)c(\022)r Fs(\))2393 2333 y Fj(\021)2462 2425 y Fp(\000)2556 2404 y Fs(^)2545 2425 y(\007)2610 2437 y Fl(I)2661 2358 y Fj(\000)2699 2425 y Fq(W)2789 2390 y Fk(+)2777 2447 y(0)2844 2425 y Fs(\()p Fq(s;)g(\022)r Fs(\))3025 2358 y Fj(\001)3064 2333 y(i)1026 2612 y Fs(=)1173 2520 y Fj(\020)1223 2612 y Fq(Y)1290 2576 y Fi(\000)1271 2640 y Fk(\(2\))1360 2520 y Fj(\021)1410 2670 y Fl(I)1462 2612 y Fs(\()p Fq(s;)g(\022)r Fs(\))19 b(+)f Fq(\026)1809 2520 y Fj(h)1860 2591 y Fs(^)1848 2612 y(\007)1913 2624 y Fl(I)1965 2520 y Fj(\020)2014 2612 y Fq(W)2104 2576 y Fi(\000)2092 2640 y Fk(\(2\))2182 2612 y Fs(\()p Fq(s;)c(\022)r Fs(\))2363 2520 y Fj(\021)2431 2612 y Fp(\000)2526 2591 y Fs(^)2514 2612 y(\007)2579 2624 y Fl(I)2631 2545 y Fj(\000)2669 2612 y Fq(W)2759 2576 y Fk(+)2747 2634 y(0)2814 2612 y Fs(\()p Fq(s;)g(\022)r Fs(\))2995 2545 y Fj(\001)3034 2520 y(i)1026 2799 y Fs(=)1173 2707 y Fj(h)q(\020)1262 2799 y Fq(Y)1329 2763 y Fi(\000)1310 2827 y Fk(\(1\))1400 2707 y Fj(\021)1449 2857 y Fl(I)1501 2799 y Fs(\()p Fq(s)19 b Fs(+)f Fq(s)1713 2811 y Fk(0)1750 2799 y Fq(;)c(\022)r Fs(\))19 b Fp(\000)f Fq(Y)2029 2763 y Fk(+)2010 2823 y Fl(I)2084 2799 y Fs(\()p Fq(s)g Fs(+)g Fq(s)2295 2811 y Fk(0)2332 2799 y Fq(;)c(\022)r Fs(\))2442 2707 y Fj(i)1173 2986 y Fs(+)p Fq(\026)1302 2893 y Fj(h)1350 2965 y Fs(^)1341 2986 y(\010)1401 2998 y Fl(I)1453 2893 y Fj(\020)1503 2986 y Fq(Y)1569 2950 y Fi(\000)1551 3014 y Fk(\(1\))1640 2986 y Fs(\()p Fq(s)19 b Fs(+)f Fq(s)1852 2998 y Fk(0)1889 2986 y Fq(;)c(\022)r Fs(\))1999 2893 y Fj(\021)2067 2986 y Fp(\000)2160 2965 y Fs(^)2150 2986 y(\010)2210 2998 y Fl(I)2262 2918 y Fj(\000)2300 2986 y Fq(Y)2367 2951 y Fk(+)2422 2986 y Fs(\()p Fq(s)19 b Fs(+)f Fq(s)2634 2998 y Fk(0)2671 2986 y Fq(;)c(\022)r Fs(\))2781 2918 y Fj(\001)2819 2893 y(i)1173 3168 y Fs(+)p Fq(\026)1302 3076 y Fj(h)1353 3147 y Fs(^)1341 3168 y(\007)1406 3180 y Fl(I)1458 3076 y Fj(\020)1507 3168 y Fq(W)1597 3133 y Fi(\000)1585 3196 y Fk(\(2\))1675 3168 y Fs(\()p Fq(s;)g(\022)r Fs(\))1856 3076 y Fj(\021)1924 3168 y Fp(\000)2019 3147 y Fs(^)2007 3168 y(\007)2072 3180 y Fl(I)2124 3101 y Fj(\000)2162 3168 y Fq(W)2252 3133 y Fk(+)2240 3190 y(0)2307 3168 y Fs(\()p Fq(s;)g(\022)r Fs(\))2488 3101 y Fj(\001)2527 3076 y(i)1026 3327 y Fs(=)1173 3260 y Fj(\002)1208 3327 y Fq(Z)1271 3291 y Fi(\000)1265 3351 y Fl(I)1327 3327 y Fs(\()p Fq(s)k Fs(+)g Fq(s)1538 3339 y Fk(0)1576 3327 y Fq(;)c(\022)r Fs(\))k Fp(\000)h Fq(Z)1851 3291 y Fk(+)1845 3351 y Fl(I)1905 3327 y Fs(\()p Fq(s)g Fs(+)f Fq(s)2117 3339 y Fk(0)2154 3327 y Fq(;)c(\022)r Fs(\))2264 3260 y Fj(\003)1173 3485 y Fs(+)p Fq(\026)1302 3392 y Fj(h)1350 3464 y Fs(^)1341 3485 y(\010)1401 3497 y Fl(I)1453 3392 y Fj(\020)1503 3485 y Fq(Y)1569 3449 y Fi(\000)1551 3513 y Fk(\(1\))1640 3485 y Fs(\()p Fq(s)19 b Fs(+)f Fq(s)1852 3497 y Fk(0)1889 3485 y Fq(;)c(\022)r Fs(\))1999 3392 y Fj(\021)2067 3485 y Fp(\000)2160 3464 y Fs(^)2150 3485 y(\010)2210 3497 y Fl(I)2262 3417 y Fj(\000)2300 3485 y Fq(Y)2367 3450 y Fk(+)2422 3485 y Fs(\()p Fq(s)19 b Fs(+)f Fq(s)2634 3497 y Fk(0)2671 3485 y Fq(;)c(\022)r Fs(\))2781 3417 y Fj(\001)2819 3392 y(i)1173 3667 y Fs(+)p Fq(\026)1302 3575 y Fj(h)1353 3646 y Fs(^)1341 3667 y(\007)1406 3679 y Fl(I)1458 3575 y Fj(\020)1507 3667 y Fq(W)1597 3632 y Fi(\000)1585 3696 y Fk(\(2\))1675 3667 y Fs(\()p Fq(s;)g(\022)r Fs(\))1856 3575 y Fj(\021)1924 3667 y Fp(\000)2019 3646 y Fs(^)2007 3667 y(\007)2072 3679 y Fl(I)2124 3600 y Fj(\000)2162 3667 y Fq(W)2252 3632 y Fk(+)2240 3689 y(0)2307 3667 y Fs(\()p Fq(s;)g(\022)r Fs(\))2488 3600 y Fj(\001)2527 3575 y(i)2580 3667 y Fq(:)-260 3892 y Fs(No)n(w)33 b(w)n(e)g(substract)f Fq(\026M)9 b Fs(\()p Fq(s)22 b Fs(+)g Fq(s)788 3904 y Fk(0)825 3892 y Fq(;)14 b(\022)r Fs(\))33 b(and)g(apply)g(Prop)r(osition)f(8,)i(and)f (b)r(ounds)g(\(64\))g(and)g(\(66\))g(together)f(with)h(the)h(second)e (ro)n(w)-260 3991 y(of)c(\(65\))f(and)g(\(67\).)37 b(W)-7 b(e)28 b(obtain)f(statemen)n(t)h(\(b\):)883 4120 y Fj(\014)883 4170 y(\014)910 4191 y Fq(W)1000 4155 y Fi(\000)988 4215 y Fl(I)1056 4191 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))19 b Fp(\000)f Fq(\026M)9 b Fs(\()p Fq(s)19 b Fs(+)f Fq(s)1691 4203 y Fk(0)1728 4191 y Fq(;)c(\022)r Fs(\))1838 4120 y Fj(\014)1838 4170 y(\014)1049 4377 y Fp(\026)1264 4321 y Fs(1)p 1146 4358 277 4 v 1146 4434 a Fq(\016)1186 4410 y Fk(2)p Fl(l)p Fk(+)p Fl(\013)p Fk(+1)1447 4260 y Fj(\022)1530 4321 y Fq(\026)1580 4291 y Fk(2)p 1518 4358 111 4 v 1518 4434 a Fq(\016)1558 4410 y Fl(p)1592 4418 y Fe(4)1657 4377 y Fs(+)1816 4321 y Fq(\026)1866 4291 y Fk(2)p 1750 4358 219 4 v 1750 4434 a Fq(\016)1790 4410 y Fl(p)1824 4418 y Fe(2)1861 4374 y Fp(p)p 1930 4374 39 4 v 60 x Fq(")1979 4260 y Fj(\023)2058 4377 y Fs(+)2264 4321 y Fq(\026)2314 4291 y Fk(2)p 2151 4358 313 4 v 2151 4434 a Fq(\016)2191 4410 y Fl(p)2225 4418 y Fe(3)2258 4410 y Fk(+)p Fl(p)2343 4418 y Fe(4)2376 4410 y Fk(+3)2492 4377 y Fs(+)2752 4321 y Fq(\026)2802 4291 y Fk(2)p 2585 4358 421 4 v 2585 4434 a Fq(\016)2625 4410 y Fl(p)2659 4418 y Fe(2)2692 4410 y Fk(+)p Fl(p)2777 4418 y Fe(3)2810 4410 y Fk(+3)2898 4374 y Fp(p)p 2967 4374 39 4 v 60 x Fq(")1049 4613 y Fp(\026)1158 4557 y Fq(\026)1208 4527 y Fk(2)p 1146 4594 111 4 v 1146 4670 a Fq(\016)1186 4646 y Fl(p)1220 4654 y Fe(6)1286 4613 y Fs(+)1444 4557 y Fq(\026)1494 4527 y Fk(2)p 1379 4594 219 4 v 1379 4670 a Fq(\016)1419 4646 y Fl(p)1453 4654 y Fe(7)1489 4610 y Fp(p)p 1559 4610 39 4 v 1559 4670 a Fq(")1607 4613 y(;)-260 4861 y Fs(since)28 b Fq(p)-14 4873 y Fk(6)46 4861 y Fs(=)22 b(max\()p Fq(p)362 4873 y Fk(4)418 4861 y Fs(+)c(2)p Fq(l)h Fs(+)f Fq(\013)h Fs(+)f(1)p Fq(;)c(p)946 4873 y Fk(3)1001 4861 y Fs(+)k Fq(p)1126 4873 y Fk(4)1181 4861 y Fs(+)g(3\))28 b(and)f Fq(p)1569 4873 y Fk(7)1629 4861 y Fs(=)c(max)o(\()p Fq(p)1945 4873 y Fk(2)2001 4861 y Fs(+)18 b(2)p Fq(l)h Fs(+)f Fq(\013)h Fs(+)f(1)p Fq(;)c(p)2529 4873 y Fk(2)2584 4861 y Fs(+)k Fq(p)2709 4873 y Fk(3)2765 4861 y Fs(+)g(3\).)-135 5060 y(Finally)-7 b(,)36 b(part)e(\(c\))g(is)h (a)e(direct)i(consequence)e(of)h(the)h(fact)f(that)h(the)f(inner)h (\015o)n(w)e(on)h Fp(W)2745 5030 y Fi(\000)2835 5060 y Fs(is)h(giv)n(en)e(b)n(y)47 b(_)-36 b Fq(s)34 b Fs(=)g Fq(b)p Fs(,)3554 5038 y(_)3538 5060 y Fq(\022)i Fs(=)41 b(~)-49 b Fq(!)3764 5072 y Fl(")3799 5060 y Fs(,)36 b(and)-260 5159 y(the)c(Hamiltonian)f(equations)g(\(35\))g(in)h(the)f(\015o)n(w-b) r(o)n(x)f(co)r(ordinates.)47 b(F)-7 b(or)31 b(instance,)h(from)g(the)f (equation)3204 5138 y(_)3185 5159 y Fq(I)37 b Fs(=)29 b(0)i(w)n(e)g(get)g(that)h(the)-260 5259 y Fq(I)7 b Fs(-comp)r(onen)n (t)27 b Fq(W)325 5224 y Fi(\000)313 5283 y Fl(I)381 5259 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))28 b(is)35 b(^)-49 b Fq(!)726 5271 y Fl(")761 5259 y Fs(-quasip)r(erio)r(dic:)644 5512 y(0)23 b(=)831 5456 y(d)p 816 5493 77 4 v 816 5569 a(d)p Fq(t)902 5392 y Fj(\014)902 5442 y(\014)902 5491 y(\014)902 5541 y(\014)930 5595 y Fl(t)p Fk(=0)1057 5512 y Fq(W)1147 5477 y Fi(\000)1135 5536 y Fl(I)1217 5512 y Fs(\()p Fq(s)c Fs(+)f Fq(bt;)c(\022)20 b Fs(+)25 b(~)-49 b Fq(!)1687 5524 y Fl(")1722 5512 y Fq(t)p Fs(\))23 b(=)g Fq(b@)1975 5524 y Fl(s)2010 5512 y Fq(W)2100 5477 y Fi(\000)2088 5536 y Fl(I)2156 5512 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))19 b(+)2439 5445 y Fj(\012)2485 5512 y Fs(~)-48 b Fq(!)2531 5524 y Fl(")2566 5512 y Fq(;)14 b(@)2647 5524 y Fl(\022)2684 5512 y Fq(W)2774 5477 y Fi(\000)2762 5536 y Fl(I)2830 5512 y Fs(\()p Fq(s;)g(\022)r Fs(\))3011 5445 y Fj(\013)3065 5512 y Fq(:)3930 5740 y Fd(2)1825 5989 y Fs(21)p eop %%Page: 22 22 22 21 bop -260 -82 a Fn(4.6)112 b(The)38 b(splitting)d(p)s(oten)m(tial) h(and)i(the)g(splitting)c(function)-260 171 y Fs(It)g(has)g(to)g(b)r(e) g(p)r(oin)n(ted)g(out)g(that)g(the)h(quasip)r(erio)r(dicit)n(y)d(of)i (the)h(function)f Fq(W)2294 135 y Fi(\000)2282 195 y Fl(I)2350 171 y Fs(\()p Fq(s;)14 b(\022)r Fs(\),)36 b(established)e(in) g(Prop)r(osition)e(9,)k(do)r(es)d(not)-260 270 y(lead)26 b(directly)g(to)g(exp)r(onen)n(tially)g(small)g(estimates)g(for)f(the)i (splitting,)g(b)r(ecause)f(an)g(essen)n(tial)f(ingredien)n(t)h(still)g (missing)g(is)g(that)h(the)-260 370 y(function)j(considered)f(has)f(to) i(ha)n(v)n(e)e(zero)g(a)n(v)n(erage)f(with)j(resp)r(ect)f(to)g(the)h (angles.)42 b(This)29 b(is)g(closely)g(related)g(to)g(the)h(main)f (result)g(of)-260 490 y([DG00],)i(whic)n(h)e(sa)n(ys)g(that,)i(after)e (a)h(suitable)g(c)n(hange)e(of)i(parameters)f(\()p Fq(s;)14 b(\022)r Fs(\))27 b(=)g Fp(A)2444 398 y Fj(\020)2497 490 y Fs(~)-46 b Fq(s;)2576 468 y Fs(~)2569 490 y Fq(\022)2611 398 y Fj(\021)2660 490 y Fs(,)31 b(the)f(function)h Fq(W)3277 454 y Fi(\000)3265 514 y Fl(I)3333 490 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))30 b(b)r(ecomes)g(the)-260 648 y(gradien)n(t)d(of)g(a)g (scalar)f(function)i Fp(L)861 555 y Fj(\020)914 648 y Fs(~)-45 b Fq(s;)994 626 y Fs(~)987 648 y Fq(\022)1028 555 y Fj(\021)1078 648 y Fs(,)27 b(called)h Fo(splitting)i(p)l (otential)p Fs(.)38 b(The)27 b(c)n(hange)g Fp(A)h Fs(is)f(de\014ned)h (as)f(the)h(in)n(v)n(erse)e(of)1278 857 y(~)-45 b Fq(s)23 b Fs(=)f Fq(W)1514 821 y Fi(\000)1502 881 y Fl(S)1570 857 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))p Fq(;)1962 835 y Fs(~)1955 857 y Fq(\022)25 b Fs(=)e Fq(W)2197 822 y Fi(\000)2185 877 y Fl(')2253 857 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))p Fq(:)1388 b Fs(\(68\))-260 1037 y(Then,)38 b(the)f(global)d (unstable)i(whisk)n(er)f Fp(W)1133 1007 y Fi(\000)1225 1037 y Fs(can)g(b)r(e)i(seen)e(as)h(a)f(graphic)g(o)n(v)n(er)f(the)i (lo)r(cal)g(stable)f(whisk)n(er)g Fp(W)3375 1002 y Fk(+)3368 1063 y(lo)r(c)3491 1037 y Fs(giv)n(en)g(in)h(\(59\).)-260 1137 y(Indeed,)28 b(in)g(the)g(new)g(parameters)d(the)j(whisk)n(er)f Fp(W)1406 1107 y Fi(\000)1489 1137 y Fs(b)r(ecomes)831 1306 y Fq(W)921 1272 y Fi(\000)977 1306 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))84 b(=)1390 1239 y Fj(\000)1428 1306 y Fq(W)1518 1271 y Fi(\000)1506 1330 y Fl(S)1574 1306 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))p Fq(;)g(W)1882 1271 y Fi(\000)1870 1330 y Fl(E)1938 1306 y Fs(\()p Fq(s;)g(\022)r Fs(\))p Fq(;)g(W)2246 1272 y Fi(\000)2234 1327 y Fl(')2303 1306 y Fs(\()p Fq(s;)g(\022)r Fs(\))p Fq(;)g(W)2611 1271 y Fi(\000)2599 1330 y Fl(I)2667 1306 y Fs(\()p Fq(s;)g(\022)r Fs(\))2848 1239 y Fj(\001)1242 1467 y Fs(=)1390 1374 y Fj(\020)1443 1467 y Fs(~)-46 b Fq(s;)14 b Fp(E)1580 1374 y Fj(\020)1633 1467 y Fs(~)-45 b Fq(s;)1712 1445 y Fs(~)1706 1467 y Fq(\022)1747 1374 y Fj(\021)1810 1467 y Fq(;)1854 1445 y Fs(~)1847 1467 y Fq(\022)r(;)14 b Fp(M)2039 1374 y Fj(\020)2092 1467 y Fs(~)-46 b Fq(s;)2171 1445 y Fs(~)2164 1467 y Fq(\022)2206 1374 y Fj(\021)o(\021)2319 1467 y Fq(:)-260 1664 y Fs(W)-7 b(e)31 b(stress)f(that)h(part)f(\(c\))h (of)f(Prop)r(osition)f(9)i(implies)f(that,)i(in)f(the)g(new)g (parameters,)e(the)i(inner)g(\015o)n(w)f(on)g Fp(W)3347 1634 y Fi(\000)3433 1664 y Fs(is)h(still)g(giv)n(en)e(b)n(y)-252 1771 y(_)-257 1790 y(~)-45 b Fq(s)29 b Fs(=)f Fq(b)p Fs(,)1 1749 y(_)-2 1769 y(~)-9 1790 y Fq(\022)j Fs(=)k(~)-48 b Fq(!)207 1802 y Fl(")242 1790 y Fs(.)47 b(T)-7 b(o)30 b(study)h(the)h(splitting,)g(w)n(e)e(only)h(need)g(to)g(consider)f(the) h(function)g Fp(M)p Fs(,)h(that)f(w)n(e)g(call)f(the)i Fo(splitting)h(function)p Fs(.)-260 1890 y(The)28 b(function)h Fp(E)35 b Fs(is)28 b(directly)f(related)h(to)f Fp(M)h Fs(b)n(y)g(the)g(energy)f(conserv)-5 b(ation,)26 b(as)h(will)h(b)r(e)h (seen)e(in)i(\(72\).)37 b(Besides,)27 b(as)h(in)g([DG00])g(w)n(e)-260 1990 y(shall)f(ha)n(v)n(e:)1429 2089 y Fp(E)j Fs(=)23 b Fq(@)1638 2101 y Fk(~)-36 b Fl(s)1671 2089 y Fp(L)p Fq(;)180 b Fp(M)23 b Fs(=)f Fq(@)2191 2099 y Fk(~)2185 2115 y Fl(\022)2223 2089 y Fp(L)p Fq(:)1542 b Fs(\(69\))-260 2261 y(So)25 b(the)g(function)h Fp(M)430 2168 y Fj(\020)482 2261 y Fs(~)-45 b Fq(s;)561 2239 y Fs(~)555 2261 y Fq(\022)596 2168 y Fj(\021)671 2261 y Fs(is)24 b(a)h(reparameterization)d(of)j(the) h(function)f Fq(W)2171 2230 y Fi(\000)2227 2261 y Fs(\()p Fq(s;)14 b(\022)r Fs(\),)26 b(b)r(ecoming)f(no)n(w)f(the)i(gradien)n(t) e(of)h(the)g(splitting)-260 2380 y(p)r(oten)n(tial.)37 b(W)-7 b(e)28 b(then)g(de\014ne)g(the)g Fo(err)l(or)i(term)d Fs(for)g(the)h(P)n(oincar)n(\023)-39 b(e{Melnik)n(o)n(v)23 b(metho)r(d)28 b(as)f(the)h(function)1153 2570 y Fp(R)1237 2478 y Fj(\020)1290 2570 y Fs(~)-45 b Fq(s;)1369 2548 y Fs(~)1363 2570 y Fq(\022)1404 2478 y Fj(\021)1476 2570 y Fs(=)23 b Fp(M)1678 2478 y Fj(\020)1730 2570 y Fs(~)-45 b Fq(s;)1810 2548 y Fs(~)1803 2570 y Fq(\022)1844 2478 y Fj(\021)1912 2570 y Fp(\000)18 b Fq(\026M)2149 2478 y Fj(\020)2202 2570 y Fs(~)-46 b Fq(s)19 b Fs(+)f Fq(s)2378 2582 y Fk(0)2415 2570 y Fq(;)2459 2548 y Fs(~)2452 2570 y Fq(\022)2493 2478 y Fj(\021)2556 2570 y Fq(:)1266 b Fs(\(70\))-260 2896 y Ft(Theorem)31 b(10)41 b Fo(Under)32 b(c)l(onditions)i(\(26\))f(and)g(\(49\),)i(the)e(splitting)g(function)g Fp(M)2406 2804 y Fj(\020)2458 2896 y Fs(~)-45 b Fq(s;)2537 2875 y Fs(~)2531 2896 y Fq(\022)2572 2804 y Fj(\021)2654 2896 y Fo(c)l(an)33 b(b)l(e)g(de\014ne)l(d)g(for)3333 2804 y Fj(\020)3386 2896 y Fs(~)-45 b Fq(s;)3465 2875 y Fs(~)3459 2896 y Fq(\022)3500 2804 y Fj(\021)3578 2896 y Fp(2)29 b(P)3720 2908 y Fl(\024)3759 2916 y Fe(3)3791 2908 y Fl(;\027)3844 2916 y Fe(3)3876 2908 y Fl(;\032)3930 2916 y Fe(6)3967 2896 y Fo(,)-260 3021 y(and)h(with)h(the)f(exp)l (onents)f Fq(p)642 3033 y Fl(i)699 3021 y Fo(de\014ne)l(d)h(in)g (\(14{19\),)i(one)e(has:)-200 3297 y Fs(\(a\))42 b Fp(jMj)94 3322 y Fl(\024)133 3330 y Fe(3)165 3322 y Fl(;\027)218 3330 y Fe(3)250 3322 y Fl(;\032)304 3330 y Fe(6)364 3297 y Fp(\026)492 3241 y Fq(\026)p 462 3278 111 4 v 462 3354 a(\016)502 3330 y Fl(p)536 3338 y Fe(4)601 3297 y Fs(+)778 3241 y Fq(\026)p 694 3278 219 4 v 694 3354 a(\016)734 3330 y Fl(p)768 3338 y Fe(2)805 3295 y Fp(p)p 874 3295 39 4 v 59 x Fq(")922 3297 y Fo(.)-204 3567 y Fs(\(b\))g Fp(jRj)64 3592 y Fl(\024)103 3600 y Fe(3)135 3592 y Fl(;\027)188 3600 y Fe(3)221 3592 y Fl(;\032)275 3600 y Fe(6)335 3567 y Fp(\026)444 3511 y Fq(\026)494 3481 y Fk(2)p 432 3548 111 4 v 432 3624 a Fq(\016)472 3600 y Fl(p)506 3608 y Fe(6)571 3567 y Fs(+)730 3511 y Fq(\026)780 3481 y Fk(2)p 664 3548 219 4 v 664 3625 a Fq(\016)704 3601 y Fl(p)738 3609 y Fe(7)775 3565 y Fp(p)p 844 3565 39 4 v 60 x Fq(")893 3567 y Fo(.)-195 3809 y Fs(\(c\))g Fp(M)22 b Fs(=)h Fq(@)202 3821 y Fl(\022)240 3809 y Fp(L)30 b Fo(for)g(a)g(sc)l(alar)h(and)37 b Fs(^)-49 b Fq(!)983 3821 y Fl(")1019 3809 y Fo(-quasip)l(erio)l(dic) 32 b(function)d Fp(L)1941 3717 y Fj(\020)1994 3809 y Fs(~)-45 b Fq(s;)2074 3787 y Fs(~)2067 3809 y Fq(\022)2108 3717 y Fj(\021)2158 3809 y Fo(.)-260 4103 y(Pr)l(o)l(of.)60 b Fs(First,)29 b(using)f(part)g(\(a\))h(of)g(Prop)r(osition)e(9,)i(w)n (e)f(see)h(that)g(the)g(c)n(hange)f(\(68\))g(is)h(close)f(to)h(the)g (iden)n(tit)n(y)-7 b(.)40 b(More)28 b(precisely)-7 b(,)29 b(for)-260 4202 y(\()p Fq(s;)14 b(\022)r Fs(\))24 b Fp(2)f(P)81 4214 y Fl(\024)120 4222 y Fe(2)152 4214 y Fl(;\027)205 4222 y Fe(2)238 4214 y Fl(;\032)292 4222 y Fe(5)328 4202 y Fs(,)939 4330 y Fp(j)t Fs(~)-46 b Fq(s)19 b Fp(\000)f Fq(s)p Fp(j)23 b(\026)1316 4274 y Fq(\026)p 1285 4311 111 4 v 1285 4387 a(\016)1325 4363 y Fl(p)1359 4371 y Fe(5)1424 4330 y Fs(+)1602 4274 y Fq(\026)p 1517 4311 219 4 v 1517 4387 a(\016)1557 4363 y Fl(p)1591 4371 y Fe(2)1628 4327 y Fp(p)p 1697 4327 39 4 v 60 x Fq(")1746 4330 y(;)1949 4234 y Fj(\014)1949 4284 y(\014)1949 4334 y(\014)1984 4308 y Fs(~)1977 4330 y Fq(\022)e 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4730 y Fe(6)1118 4710 y Fs(.)68 b(Then,)41 b(part)d(\(a\))g(is)g(ob)n (vious)f(from)h(part)f(\(a\))h(of)g(Prop)r(osition)f(9)g(\(taking)h (only)g(the)-260 4835 y Fq(I)7 b Fs(-comp)r(onen)n(t\).)-135 5034 y(T)-7 b(o)27 b(establish)g(part)g(\(b\),)i(w)n(e)e(write)498 5228 y Fp(R)582 5136 y Fj(\020)635 5228 y Fs(~)-45 b Fq(s)o(;)714 5206 y Fs(~)707 5228 y Fq(\022)749 5136 y Fj(\021)821 5228 y Fs(=)909 5161 y Fj(\002)944 5228 y Fq(W)1034 5192 y Fi(\000)1022 5252 y Fl(I)1090 5228 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))19 b Fp(\000)f Fq(\026M)9 b Fs(\()p Fq(s)18 b Fs(+)g Fq(s)1724 5240 y Fk(0)1761 5228 y Fq(;)c(\022)r Fs(\))1871 5161 y Fj(\003)1925 5228 y Fs(+)k Fq(\026)2072 5136 y Fj(h)2111 5228 y Fq(M)9 b Fs(\()p Fq(s)18 b Fs(+)g Fq(s)2412 5240 y Fk(0)2449 5228 y Fq(;)c(\022)r Fs(\))19 b Fp(\000)f Fq(M)2764 5136 y Fj(\020)2817 5228 y Fs(~)-45 b Fq(s)18 b Fs(+)g Fq(s)2993 5240 y Fk(0)3031 5228 y Fq(;)3074 5206 y Fs(~)3068 5228 y Fq(\022)3109 5136 y Fj(\021)o(i)3211 5228 y Fq(:)-260 5422 y Fs(The)32 b(most)g(signi\014can)n(t)g(term)g(in)g(this)h(sum)f (is)g(the)g(\014rst)g(one,)h(and)f(has)g(b)r(een)g(b)r(ounded)h(in)f (part)g(\(b\))h(of)f(Prop)r(osition)e(9.)50 b(F)-7 b(or)32 b(the)-260 5521 y(second)27 b(term,)h(w)n(e)f(deduce)h(from)f(Lemma)g (7)h(the)f(b)r(ounds)851 5725 y Fp(j)p Fq(@)918 5737 y Fl(s)954 5725 y Fq(M)1035 5737 y Fl(s)1066 5745 y Fe(0)1102 5725 y Fp(j)1125 5750 y Fl(\024)1164 5758 y Fe(1)1196 5750 y Fl(;\027)1249 5758 y Fe(2)1282 5750 y Fl(;\032)1336 5758 y Fe(2)1395 5725 y Fp(\026)1611 5669 y Fs(1)p 1493 5706 277 4 v 1493 5782 a Fq(\016)1533 5758 y Fk(2)p Fl(l)p Fk(+)p Fl(\013)p Fk(+1)1780 5725 y Fq(;)180 b Fp(j)p Fq(@)2050 5737 y Fl(\022)2087 5725 y Fq(M)2168 5737 y Fl(s)2199 5745 y Fe(0)2236 5725 y Fp(j)2259 5750 y Fl(\024;\027)2351 5758 y Fe(1)2383 5750 y Fl(;\032)2437 5758 y Fe(3)2497 5725 y Fp(\026)2712 5669 y Fs(1)p 2595 5706 V 2595 5782 a Fq(\016)2635 5758 y Fk(2)p Fl(l)p Fk(+)p Fl(\013)p Fk(+1)1825 5989 y Fs(22)p eop %%Page: 23 23 23 22 bop -260 -82 a Fs(and,)28 b(using)f(also)f(\(71\),)i(w)n(e)f (obtain)861 30 y Fj(\014)861 80 y(\014)861 130 y(\014)889 125 y Fq(M)9 b Fs(\()p Fq(s)18 b Fs(+)g Fq(s)1190 137 y Fk(0)1227 125 y Fq(;)c(\022)r Fs(\))19 b Fp(\000)f Fq(M)1542 33 y Fj(\020)1595 125 y Fs(~)-45 b Fq(s)18 b Fs(+)g Fq(s)1771 137 y Fk(0)1809 125 y Fq(;)1852 103 y Fs(~)1846 125 y Fq(\022)1887 33 y Fj(\021)1936 30 y(\014)1936 80 y(\014)1936 130 y(\014)1027 333 y Fp(\026)1242 277 y Fs(1)p 1125 314 277 4 v 1125 390 a Fq(\016)1165 366 y Fk(2)p Fl(l)p Fk(+)p Fl(\013)p Fk(+1)1425 216 y Fj(\022)1527 277 y Fq(\026)p 1496 314 111 4 v 1496 390 a(\016)1536 366 y Fl(p)1570 374 y Fe(5)1636 333 y Fs(+)1813 277 y Fq(\026)p 1729 314 219 4 v 1729 390 a(\016)1769 366 y Fl(p)1803 374 y Fe(2)1839 330 y Fp(p)p 1909 330 39 4 v 1909 390 a Fq(")1957 216 y Fj(\023)2037 333 y Fs(+)2247 277 y(1)p 2130 314 277 4 v 2130 390 a Fq(\016)2170 366 y Fk(2)p Fl(l)p Fk(+)p Fl(\013)p Fk(+1)2430 216 y Fj(\022)2532 277 y Fq(\026)p 2501 314 111 4 v 2501 390 a(\016)2541 366 y Fl(p)2575 374 y Fe(4)2641 333 y Fs(+)2818 277 y Fq(\026)p 2734 314 219 4 v 2734 390 a(\016)2774 366 y Fl(p)2808 374 y Fe(2)2844 330 y Fp(p)p 2914 330 39 4 v 2914 390 a Fq(")2962 216 y Fj(\023)1027 565 y Fp(\026)1242 509 y Fs(1)p 1125 546 277 4 v 1125 622 a Fq(\016)1165 598 y Fk(2)p Fl(l)p Fk(+)p Fl(\013)p Fk(+1)1425 448 y Fj(\022)1527 509 y Fq(\026)p 1496 546 111 4 v 1496 622 a(\016)1536 598 y Fl(p)1570 606 y Fe(5)1636 565 y Fs(+)1813 509 y Fq(\026)p 1729 546 219 4 v 1729 623 a(\016)1769 599 y Fl(p)1803 607 y Fe(2)1839 563 y Fp(p)p 1909 563 39 4 v 1909 623 a Fq(")1957 448 y Fj(\023)2032 565 y Fq(:)-260 798 y Fs(Putting)26 b(this)h(b)r(ound)g(and)f(the)h(one)f(of) g(part)g(\(b\))h(of)f(Prop)r(osition)f(9)h(together,)g(from)g(the)g (inequalit)n(y)g Fq(p)3041 810 y Fk(5)3094 798 y Fs(+)16 b(2)p Fq(l)h Fs(+)e Fq(\013)h Fs(+)g(1)23 b Fp(\024)f Fq(p)3683 810 y Fk(6)3747 798 y Fs(w)n(e)k(can)-260 898 y(k)n(eep)h Fq(p)-29 910 y Fk(6)36 898 y Fs(as)g(in)h(\(18\))f(and)g (obtain)h(the)g(exp)r(ected)g(b)r(ound)g(for)f Fp(jRj)p Fs(.)-135 1122 y(Concerning)e(part)h(\(c\),)i(the)f(existence)f(of)h (the)g(splitting)g(p)r(oten)n(tial)f Fp(L)2110 1030 y Fj(\020)2163 1122 y Fs(~)-45 b Fq(s;)2243 1100 y Fs(~)2236 1122 y Fq(\022)2277 1030 y Fj(\021)2353 1122 y Fs(w)n(as)26 b(established)g(in)h([DG00],)g(from)f(the)h(fact)g(that)-260 1246 y(there)d(exists)f(an)g(exact)g(symplectic)h(map)f(\002)g(\(close) g(to)h(the)g(iden)n(tit)n(y\),)h(de\014ned)e(in)h(a)f(neigh)n(b)r (ourho)r(o)r(d)g(of)g(a)h(piece)f(of)h(the)g(lo)r(cal)e(stable)-260 1346 y(whisk)n(er)28 b Fp(W)131 1310 y Fk(+)124 1371 y(lo)r(c)211 1346 y Fs(,)h(taking)g(this)g(whisk)n(er)e(in)n(to)i(the)g (global)f(unstable)h(one,)f Fp(W)2144 1316 y Fi(\000)2200 1346 y Fs(.)41 b(In)29 b(fact,)g(the)g(parameterizations)e(of)i(these)g (whisk)n(ers)-260 1445 y(are)36 b(link)n(ed)g(b)n(y)g(this)h(map:)55 b(\002)14 b(\()o Fq(Z)844 1415 y Fk(+)899 1445 y Fs(\()p Fq(s)19 b Fs(+)f Fq(s)1111 1457 y Fk(0)1148 1445 y Fq(;)c(\022)r Fs(\)\))39 b(=)e Fq(Z)1494 1415 y Fi(\000)1550 1445 y Fs(\()p Fq(s)24 b Fs(+)g Fq(s)1773 1457 y Fk(0)1811 1445 y Fq(;)14 b(\022)r Fs(\).)64 b(This)36 b(map)h(can)f(b)r(e)h(mo)n(v)n (ed)e(to)i(the)g(\015o)n(w-b)r(o)n(x)e(co)r(ordinates:)-251 1534 y Fj(b)-260 1555 y Fs(\002)24 b(=)g(\011)-17 1525 y Fi(\000)p Fk(1)91 1555 y Fp(\016)18 b Fs(\002)h Fp(\016)f Fs(\011,)29 b(and)f(hence)815 1534 y Fj(b)805 1555 y Fs(\002)884 1488 y Fj(\000)922 1555 y Fq(W)1012 1520 y Fk(+)1000 1578 y(0)1067 1555 y Fs(\()p Fq(s;)14 b(\022)r Fs(\))1248 1488 y Fj(\001)1311 1555 y Fs(=)24 b Fq(W)1490 1525 y Fi(\000)1546 1555 y Fs(\()p Fq(s;)14 b(\022)r Fs(\).)40 b(Nev)n(ertheless,)28 b(there)g(is)g(a)g(small)g (di\013erence)h(b)r(et)n(w)n(een)f([DG00])g(and)-260 1655 y(this)33 b(pap)r(er.)50 b(Indeed,)34 b(in)f(this)f(pap)r(er)g (one)g(of)g(the)h(ingredien)n(ts)e(of)i(\011)f(is)g(not)g(exact)g(in)g (general,)h(since)f(the)g(transformation)f(\010)h(to)-260 1755 y(normal)25 b(form)g(\(pro)n(vided)g(b)n(y)g(Theorem)g(1\))g(is)g (not)h(exact,)g(but)g(rather)e(a)h(translation)g(of)g(it,)i(\010)14 b Fp(\016)g Fq(T)2886 1725 y Fi(\000)p Fk(1)2874 1775 y Fl(a)2974 1755 y Fs(,)26 b(is)f(exact.)36 b(But)26 b(it)g(is)f(not)h(hard)-260 1865 y(to)h(see)g(that,)h(ev)n(en)f(with)h (this)f(remark,)f(the)i(map)1361 1844 y Fj(b)1351 1865 y Fs(\002)f(is)g(exact.)37 b(Once)27 b(this)g(is)h(established,)f(the)g (existence)g(of)h(a)f(splitting)g(p)r(oten)n(tial)-260 1985 y Fp(L)-189 1892 y Fj(\020)-136 1985 y Fs(~)-45 b Fq(s;)-57 1963 y Fs(~)-63 1985 y Fq(\022)-22 1892 y Fj(\021)28 1985 y Fs(,)22 b(satisfying)e(\(69\),)i(follo)n(ws)e(as)g (in)h([DG00].)35 b(Its)27 b(^)-48 b Fq(!)1564 1997 y Fl(")1599 1985 y Fs(-quasip)r(erio)r(dicit)n(y)19 b(is)i(a)g (consequence)e(of)i(the)h(Hamilton{Jacobi)d(equation)-260 2105 y(and)28 b(the)g(fact)f(that)h(b)r(oth)g(whisk)n(ers)e(b)r(elong)i (to)f(the)h(same)f(energy)g(lev)n(el:)712 2308 y(0)c(=)g Fq(K)936 2320 y Fk(0)986 2215 y Fj(\020)1039 2308 y Fs(~)-45 b Fq(s;)14 b Fs(0)p Fq(;)1197 2286 y Fs(~)1191 2308 y Fq(\022)q(;)g Fs(0)1310 2215 y Fj(\021)1382 2308 y Fs(=)23 b Fq(K)1541 2320 y Fk(0)1591 2215 y Fj(\020)1644 2308 y Fs(~)-45 b Fq(s;)14 b Fp(E)1782 2215 y Fj(\020)1835 2308 y Fs(~)-45 b Fq(s)o(;)1914 2286 y Fs(~)1907 2308 y Fq(\022)1949 2215 y Fj(\021)2012 2308 y Fq(;)2056 2286 y Fs(~)2049 2308 y Fq(\022)r(;)14 b Fp(M)2241 2215 y Fj(\020)2294 2308 y Fs(~)-46 b Fq(s;)2373 2286 y Fs(~)2366 2308 y Fq(\022)2407 2215 y Fj(\021\021)878 2490 y Fs(=)23 b Fq(b)p Fp(E)1067 2398 y Fj(\020)1120 2490 y Fs(~)-46 b Fq(s;)1199 2468 y Fs(~)1192 2490 y Fq(\022)1233 2398 y Fj(\021)1301 2490 y Fs(+)1384 2398 y Fj(D)1442 2490 y Fs(~)d Fq(!)1487 2502 y Fl(")1522 2490 y Fq(;)14 b Fp(M)1673 2398 y Fj(\020)1726 2490 y Fs(~)-46 b Fq(s;)1805 2468 y Fs(~)1798 2490 y Fq(\022)1839 2398 y Fj(\021E)1963 2490 y Fs(=)23 b Fq(b@)2134 2502 y Fk(~)-36 b Fl(s)2166 2490 y Fp(L)2237 2398 y Fj(\020)2290 2490 y Fs(~)-46 b Fq(s;)2369 2468 y Fs(~)2362 2490 y Fq(\022)2403 2398 y Fj(\021)2472 2490 y Fs(+)2555 2398 y Fj(D)2612 2490 y Fs(~)d Fq(!)2657 2502 y Fl(")2692 2490 y Fq(;)14 b(@)2779 2500 y Fk(~)2773 2516 y Fl(\022)2811 2490 y Fp(L)2882 2398 y Fj(\020)2935 2490 y Fs(~)-45 b Fq(s;)3014 2468 y Fs(~)3008 2490 y Fq(\022)3049 2398 y Fj(\021)o(E)3163 2490 y Fq(:)659 b Fs(\(72\))3930 2673 y Fd(2)-260 2872 y Ft(Remark.)70 b Fs(It)36 b(is)f(an)h(ob)n(vious)e(consequence)h(of)g (part)g(\(c\))h(that)g(the)g(error)e(term)h Fp(R)h Fs(is)g(also)41 b(^)-49 b Fq(!)2875 2884 y Fl(")2911 2872 y Fs(-quasip)r(erio)r(dic,)36 b(since)f(b)r(oth)h Fp(M)-260 2972 y Fs(and)e Fq(M)43 b Fs(ha)n(v)n(e)33 b(the)i(same)f(quasip)r(erio)r(dicit)n(y)f (parameter.)55 b(This)35 b(has)e(motiv)-5 b(ated)35 b(de\014nition)f (\(50\).)57 b(It)35 b(is)f(also)f(clear)g(that)i Fp(R)f Fs(is)h(a)-260 3071 y(gradien)n(t,)28 b(b)r(ecause)g(so)g(are)f Fp(M)h Fs(and)h Fq(M)9 b Fs(.)39 b(These)28 b(facts)h(are)e(used)i(in)f 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Fs(\(0\))p Fq(e)1967 4959 y Fi(\000)p Fl(is)p Fi(h)q Fl(k)q(;!)2199 4967 y Ff(")2231 4959 y Fi(i)2262 4993 y Fq(:)1560 b Fs(\(73\))-260 5142 y(In)28 b(particular,)e(w)n(e)i(see)f(that)h(the)g (a)n(v)n(erage)c(v)-5 b(alue)28 b Fq(g)1382 5154 y Fk(0)1446 5142 y Fs(do)r(es)g(not)f(dep)r(end)h(on)g Fq(s)p Fs(.)-135 5342 y(W)-7 b(e)19 b(next)h(pro)n(vide)e(a)g(result)h(concerning)f(the) i(exp)r(onen)n(tial)e(smallness)g(in)i Fq(")f Fs(of)g(an)g Fq(!)2475 5354 y Fl(")2510 5342 y Fs(-quasip)r(erio)r(dic)f(function.) 35 b(This)19 b(is)g(no)n(w)n(ada)n(ys)-260 5441 y(standard)29 b(\(similar)h(results)g(are)f(giv)n(en)g(in)i([DGJS97)o(,)f(Sau01)o (]\),)h(but)g(w)n(e)f(include)h(here)e(the)i(pro)r(of)e(for)h(the)h (sak)n(e)e(of)h(completeness.)-260 5541 y(Recall)22 b(that)h(the)g(v)n (ector)f Fq(!)595 5553 y Fl(")653 5541 y Fs(is)g(de\014ned)h(in)g (\(4\))g(in)g(terms)g(of)f(the)h(initial)g(frequency)f(v)n(ector)g Fq(!)s Fs(,)h(assumed)f(to)h(satisfy)f(the)h(Diophan)n(tine)-260 5641 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Fq(\021)s(;)14 b(J)8 b Fs(\))29 b(using)f(\(81\))g(and)h(\(83\).)39 b(In)28 b(order)g(to)g(pro)r(ceed)g (w)n(e)g(need)g(the)h(follo)n(wing)-260 5253 y(tec)n(hnical)c(lemma,)h (whose)f(pro)r(of)g(is)g(straighforw)n(ard,)e(using)j(h)n(yp)r(otheses) e(\(H3{H4\))i(of)f(Section)h(1.1,)f(T)-7 b(a)n(ylor)23 b(form)n(ula)i(and)g(Cauc)n(h)n(y)-260 5353 y(estimates)i(\(for)h(more) f(details,)g(see)g([DGJS97]\):)-260 5641 y Ft(Lemma)j(14)41 b Fo(F)-6 b(or)27 b Fs(0)22 b Fp(\024)h Fs(Im)14 b Fq(s)23 b Fp(\024)g Fq(\027)824 5653 y Fk(1)884 5641 y Fs(=)g Fq(\031)s(=)p Fs(2)12 b Fp(\000)g Fq(\016)s Fo(,)27 b Fp(j)14 b Fs(Im)g Fq(\022)r Fp(j)24 b(\024)e Fq(\032)1655 5653 y Fk(3)1716 5641 y Fs(=)g Fq(\032)12 b Fp(\000)g Fs(3)p Fq(\016)30 b Fo(and)e Fq(\034)k Fs(=)23 b Fp(j)p Fq(t)c Fs(+)f Fq(s)g Fp(\000)g Fq(\031)s(i=)p Fs(2)p Fp(j)k(\026)h Fq(T)12 b Fo(,)27 b(the)g(fol)t(lowing)i(b)l(ounds)e (hold,)-260 5740 y(wher)l(e)k Fq(l)g Fo(and)f Fq(\013)g Fo(ar)l(e)g(de\014ne)l(d)g(in)g(hyp)l(otheses)h(\(H3{H4\):)1825 5989 y Fs(26)p eop %%Page: 27 27 27 26 bop -200 -56 a Fs(\(a\))42 b Fp(j)p Fs(\011\()p Fq(t)18 b Fs(+)g Fq(s)p Fs(\))p Fp(j)23 b(\026)416 -112 y Fs(1)p 414 -75 46 4 v 414 1 a Fq(\034)470 -56 y(;)98 b Fp(j)p Fs(\010\()p Fq(t)19 b Fs(+)f Fq(s)p Fs(\))p Fp(j)24 b(\026)e Fq(\034)1088 -90 y Fk(2)1126 -56 y Fq(;)99 b Fp(j)p Fs(\011)1336 -90 y Fi(0)1359 -56 y Fs(\()p Fq(t)18 b Fs(+)h Fq(s)p Fs(\))p Fp(j)k(\026)1758 -112 y Fs(1)p 1737 -75 83 4 v 1737 1 a Fq(\034)1782 -23 y Fk(2)1830 -56 y Fq(;)99 b Fp(j)p Fs(\010)2035 -90 y Fi(0)2058 -56 y Fs(\()p Fq(t)19 b Fs(+)f Fq(s)p Fs(\))p Fp(j)23 b(\026)g Fq(\034)9 b Fo(.)-204 181 y Fs(\(b\))42 b Fp(j)p Fq(h)19 147 y Fk(\()p Fl(j)s Fk(\))105 181 y Fs(\()p Fq(x)184 193 y Fk(0)222 181 y Fs(\()p Fq(t)19 b Fs(+)f Fq(s)p Fs(\)\))p Fp(j)24 b(\026)665 125 y Fs(1)p 633 162 105 4 v 633 238 a Fq(\034)678 214 y Fk(2)p Fl(l)748 181 y Fo(,)30 b(wher)l(e)g Fq(h)1085 151 y Fk(\()p Fl(j)s Fk(\))1202 181 y Fo(me)l(ans)f(the)h Fq(j)5 b Fo(-derivative)32 b(of)e Fq(h)p Fs(\()p Fq(x)p Fs(\))p Fo(.)-195 418 y Fs(\(c\))42 b Fp(j)p Fq(f)9 b Fs(\()p Fq(\022)20 b Fs(+)e Fq(!)247 430 y Fl(")283 418 y Fq(t)p Fs(\))p Fp(j)23 b(\026)511 362 y Fs(1)p 489 399 88 4 v 489 475 a Fq(\016)529 451 y Fl(\013)586 418 y Fq(;)99 b Fp(j)p Fq(@)775 430 y Fl(')823 418 y Fq(f)9 b Fs(\()p Fq(\022)20 b Fs(+)e Fq(!)1099 430 y Fl(")1134 418 y Fq(t)p Fs(\))p Fp(j)24 b(\026)1405 362 y Fs(1)p 1340 399 172 4 v 1340 475 a Fq(\016)1380 451 y Fl(\013)p Fk(+1)1522 418 y Fq(;)99 b Fp(j)p Fq(@)1727 384 y Fk(2)1711 438 y Fl(')1764 418 y Fq(f)9 b Fs(\()p Fq(\022)20 b Fs(+)f Fq(!)2041 430 y Fl(")2076 418 y Fq(t)p Fs(\))p Fp(j)k(\026)2347 362 y Fs(1)p 2282 399 V 2282 475 a Fq(\016)2322 451 y Fl(\013)p Fk(+2)2463 418 y Fo(.)-204 615 y Fs(\(d\))42 b Fo(If)30 b Fp(j)p Fq(\030)94 627 y Fl(j)129 615 y Fs(\()p Fq(t)p Fs(\))p Fp(j)24 b(\026)f Fq(\025=\034)493 585 y Fl(\014)561 615 y Fp(\024)g Fs(1)29 b Fo(and)h Fp(j)p Fq(\020)940 627 y Fl(j)975 615 y Fs(\()p Fq(t)p Fs(\))p Fp(j)24 b(\026)f Fs(\012)g Fp(\026)f Fq(\016)s Fo(,)31 b Fq(j)d Fs(=)22 b(1)p Fq(;)14 b Fs(2)p Fo(,)29 b(then)h(the)g(function)g Fq(g)i Fo(de\014ne)l(d)e(in)g(\(84\))g(veri\014es)664 841 y Fp(j)p Fq(g)s Fs(\()p Fq(\030)798 853 y Fk(1)836 841 y Fs(\()p Fq(t)p Fs(\))p Fq(;)14 b(\020)1003 853 y Fk(1)1041 841 y Fs(\()p Fq(t)p Fs(\))p Fq(;)g(t)p Fs(\))19 b Fp(\000)f Fq(g)s Fs(\()p Fq(\030)1447 853 y Fk(2)1484 841 y Fs(\()p Fq(t)p Fs(\))p Fq(;)c(\020)1651 853 y Fk(2)1689 841 y Fs(\()p Fq(t)p Fs(\))p Fq(;)g(t)p Fs(\))p Fp(j)84 b(\026)2137 724 y Fj(\022)2271 785 y Fq(\025)p 2208 822 175 4 v 2208 898 a(\034)2253 874 y Fl(\014)s Fk(+2)2411 841 y Fs(+)2504 785 y Fq(\026\016)2594 755 y Fi(\000)p Fl(\013)p 2504 822 190 4 v 2546 898 a Fq(\034)2591 874 y Fk(2)p Fl(l)2703 724 y Fj(\023)2778 841 y Fp(j)p Fq(\030)2837 853 y Fk(1)2875 841 y Fs(\()p Fq(t)p Fs(\))19 b Fp(\000)f Fq(\030)3107 853 y Fk(2)3144 841 y Fs(\()p Fq(t)p Fs(\))p Fp(j)2137 1083 y Fs(+)2212 1027 y Fq(\026\016)2302 997 y Fi(\000)p Fk(\()p Fl(\013)p Fk(+1\))p 2211 1064 326 4 v 2322 1140 a Fq(\034)2367 1116 y Fk(2)p Fl(l)2560 1083 y Fp(j)p Fq(\020)2619 1095 y Fk(1)2657 1083 y Fs(\()p Fq(t)p Fs(\))h Fp(\000)f Fq(\020)2889 1095 y Fk(2)2927 1083 y Fs(\()p Fq(t)p Fs(\))p Fp(j)c Fq(;)-52 1282 y Fo(and)30 b(the)g(function)g Fq(N)38 b Fo(de\014ne)l(d)30 b(in)g(\(85\))g(veri\014es:)728 1513 y Fp(j)p Fq(N)9 b Fs(\()p Fq(\030)895 1525 y Fk(1)933 1513 y Fs(\()p Fq(t)p Fs(\))p Fq(;)14 b(\020)1100 1525 y Fk(1)1138 1513 y Fs(\()p Fq(t)p Fs(\))p Fq(;)g(t)p Fs(\))19 b Fp(\000)f Fq(N)9 b Fs(\()p Fq(\030)1577 1525 y Fk(2)1615 1513 y Fs(\()p Fq(t)p Fs(\))p Fq(;)14 b(\020)1782 1525 y Fk(2)1820 1513 y Fs(\()p Fq(t)p Fs(\))p Fq(;)g(t)p Fs(\))p Fp(j)84 b(\026)2277 1457 y Fq(\026\016)2367 1426 y Fi(\000)p Fk(\()p Fl(\013)p Fk(+1\))p 2277 1494 V 2388 1570 a Fq(\034)2433 1546 y Fk(2)p Fl(l)2626 1513 y Fp(j)p Fq(\030)2685 1525 y Fk(1)2723 1513 y Fs(\()p Fq(t)p Fs(\))19 b Fp(\000)f Fq(\030)2955 1525 y Fk(2)2992 1513 y Fs(\()p Fq(t)p Fs(\))p Fp(j)2267 1733 y Fs(+)2342 1677 y Fq(\026\016)2432 1647 y Fi(\000)p Fk(\()p Fl(\013)p Fk(+2\))p 2342 1714 V 2452 1790 a Fq(\034)2497 1766 y Fk(2)p Fl(l)2691 1733 y Fp(j)p Fq(\020)2750 1745 y Fk(1)2787 1733 y Fs(\()p Fq(t)p Fs(\))h Fp(\000)f Fq(\020)3019 1745 y Fk(2)3057 1733 y Fs(\()p Fq(t)p Fs(\))p Fp(j)c Fq(:)-135 2035 y Fs(The)34 b(pro)r(of)g(of)g (Theorem)g(13,)h(for)f(the)g(momen)n(t)g(for)g Fq(s)g Fp(2)h Fq(D)1845 2005 y Fk(+)1935 2035 y Fs(=)e Fp(f)p Fq(s)h Fp(2)h Fm(C)55 b Fs(:)34 b(0)g Fp(\024)g Fs(Im)14 b Fq(s)34 b Fp(\024)g Fq(\031)s(=)p Fs(2)22 b Fp(\000)h Fq(\016)s Fp(g)p Fs(,)35 b(is)f(based)g(on)g(the)h(next)-260 2135 y(Prop)r(ositions)23 b(16)g(and)i(17.)35 b(In)24 b(the)h(\014rst)g(one,)f(the)h(solutions)f(of)h(system)f(\(78\))g(with) h(initial)g(conditions)f(v)n(erifying)f(b)r(ounds)i(\(48\),)g(will)-260 2235 y(b)r(e)30 b(extended)g(up)g(to)f(some)g(time)h Fq(t)c Fs(=)g Fp(\000)p Fq(t)1078 2247 y Fk(1)1114 2235 y Fs(\()p Fq(s)p Fs(\))k(de\014ned)g(b)r(ello)n(w.)42 b(In)30 b(the)g(second)f(prop)r(osition,)g(w)n(e)g(tak)n(e)g Fq(t)d Fs(=)f Fp(\000)p Fq(t)3361 2247 y Fk(1)3398 2235 y Fs(\()p Fq(s)p Fs(\))30 b(as)f(the)h(initial)-260 2334 y(time.)-135 2533 y(W)-7 b(e)28 b(divide)f(the)h(complex)g(strip)f Fq(D)988 2503 y Fk(+)1071 2533 y Fs(in)h(t)n(w)n(o)f(parts)1008 2705 y Fq(D)1079 2670 y Fk(up)1239 2705 y Fs(=)83 b Fp(f)p Fq(s)22 b Fp(2)i Fm(C)44 b Fs(:)23 b Fq(\031)s(=)p Fs(2)17 b Fp(\000)i Fq(\016)1973 2670 y Fk(2)p Fl(=)p Fk(3)2100 2705 y Fp(\024)k Fs(Im)14 b Fq(s)23 b Fp(\024)f Fq(\031)s(=)p Fs(2)c Fp(\000)g Fq(\016)s Fp(g)p Fq(;)929 2829 y(D)1000 2795 y Fk(do)n(wn)1239 2829 y Fs(=)83 b Fp(f)p Fq(s)22 b Fp(2)i Fm(C)44 b Fs(:)23 b(0)f Fp(\024)h Fs(Im)14 b Fq(s)23 b Fp(\024)g Fq(\031)s(=)p Fs(2)17 b Fp(\000)h Fq(\016)2387 2795 y Fk(2)p Fl(=)p Fk(3)2492 2829 y Fp(g)-260 3000 y Fs(and)28 b(de\014ne)f(the)h(separation)e(p)r(oin)n(t)i Fp(\000)p Fq(t)998 3012 y Fk(1)1035 3000 y Fs(\()p Fq(s)p Fs(\))g(b)n(y)1045 3224 y Fp(\000)p Fq(t)1140 3236 y Fk(1)1177 3224 y Fs(\()p Fq(s)p Fs(\))19 b(+)f(Re)c Fq(s)23 b Fs(=)1644 3107 y Fj(\032)1747 3175 y Fp(\000)p Fq(\016)1852 3144 y Fk(2)p Fl(=)p Fk(3)1956 3175 y Fq(;)83 b Fs(for)g Fq(s)23 b Fp(2)g Fq(D)2456 3144 y Fk(up)2534 3175 y Fq(;)1747 3275 y Fs(0)p Fq(;)250 b Fs(for)83 b Fq(s)23 b Fp(2)g Fq(D)2456 3245 y Fk(do)n(wn)2612 3275 y Fq(:)-135 3547 y Fs(During)k(this)h(pro)r(of)f(w)n(e)g(will)h(use)g(the)f(follo)n (wing)g(tec)n(hnical)g(result)h(\(an)f(analogous)e(one)j(is)f(pro)n(v)n (ed)f(in)i([DS92]\).)-260 3817 y Ft(Lemma)i(15)41 b Fo(L)l(et)29 b Fq(t)p Fo(,)h Fq(t)474 3829 y Fk(0)541 3817 y Fo(r)l(e)l(al,)h Fq(s)e Fo(c)l(omplex,)i(such)f(that)933 3989 y Fp(j)p Fs(Im)14 b Fq(s)p Fp(j)23 b Fq(<)g(\031)s(=)p Fs(2)p Fq(;)98 b(q)1534 4001 y Fk(2)1594 3989 y Fp(\024)23 b Fs(Re)14 b Fq(s)23 b Fp(\024)f Fs(2)p Fq(q)2022 4001 y Fk(2)2059 3989 y Fq(;)99 b Fp(\000)p Fq(T)34 b Fp(\024)22 b Fq(t)h Fp(\024)g Fq(t)2587 4001 y Fk(0)2647 3989 y Fp(\024)g Fs(0)p 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Fj(\022)2028 5703 y Fs(1)p Fq(;)2117 5647 y Fs(1)p 2117 5684 42 4 v 2118 5760 a Fq(\016)2168 5586 y Fj(\023)2243 5703 y Fq(:)1579 b Fs(\(86\))1825 5989 y(27)p eop %%Page: 28 28 28 27 bop -260 -82 a Fo(Then,)36 b(under)e(the)g(hyp)l(otheses)i(of)e (The)l(or)l(em)h(13,)h Fq(u)p Fs(\()p Fq(t)p Fs(\))e Fo(c)l(an)g(b)l(e)g(extende)l(d)g(for)h Fq(t)30 b Fp(2)h Fs([)p Fp(\000)p Fq(t)2559 -70 y Fk(1)2596 -82 y Fs(\()p Fq(s)p Fs(\))p Fq(;)14 b Fs(0])34 b Fo(and)h(satis\014es)f(ther)l(e)g (the)g(fol)t(lowing)-260 17 y(b)l(ound:)730 46 y Fj(\014)730 96 y(\014)758 50 y(\000)796 117 y Fq(\034)9 b(\030)t Fs(\()p Fq(t)p Fs(\))p Fq(;)14 b(\034)1057 83 y Fk(2)1096 117 y Fq(\021)s Fs(\()p Fq(t)p Fs(\))p Fq(;)g(\020)6 b Fs(\()p Fq(t)p Fs(\))p Fq(;)14 b(J)8 b Fs(\()p Fq(t)p Fs(\))1592 50 y Fj(\001)1632 46 y(\014)1632 96 y(\014)1683 117 y Fp(\026)1771 25 y Fj(\020)1860 61 y Fq(\026)p 1830 98 111 4 v 1830 174 a(\016)1870 150 y Fl(p)1904 158 y Fe(3)1969 117 y Fs(+)2176 61 y Fq(\026)p 2062 98 278 4 v 2062 174 a(\016)2102 150 y Fk(2)p Fl(l)p Fk(+)p Fl(\013)p Fi(\000)p Fk(2)2350 117 y Fq(;)2469 61 y(\026)p 2397 98 195 4 v 2397 174 a(\016)2437 150 y Fl(p)2471 158 y Fe(3)2503 150 y Fk(+1)2620 117 y Fs(+)2784 61 y Fq(\026)p 2713 98 193 4 v 2713 174 a(\016)2753 150 y Fk(2)p Fl(l)p Fk(+)p Fl(\013)2916 25 y Fj(\021)2979 117 y Fq(:)843 b Fs(\(87\))-260 419 y Fo(Pr)l(o)l(of.)49 b Fs(W)-7 b(e)25 b(shall)e(use)h(the)g(metho)r(d)g(of)g(successiv)n(e)e(appro)n (ximations.)34 b(W)-7 b(e)24 b(b)r(egin)g(the)g(iteration)f(pro)r(cess) g(with)h Fq(\030)3340 431 y Fk(0)3377 419 y Fs(\()p Fq(t)p Fs(\))g(=)f(0,)h Fq(\020)3708 431 y Fk(0)3745 419 y Fs(\()p Fq(t)p Fs(\))g(=)f(0)-260 519 y(and)28 b(consider,)e(for)h Fq(n)c Fp(\025)g Fs(0,)k(the)h(recurrence)e(suggested)h(b)n(y)g(the)h (system)f(of)h(equations)e(\(80\),)i(\(82\):)495 751 y Fq(\030)531 763 y Fl(n)p Fk(+1)661 751 y Fs(\()p Fq(t)p Fs(\))23 b(=)g(\011\()p Fq(t)18 b Fs(+)g Fq(s)p Fs(\))1179 634 y Fj(\024)1223 751 y Fs(\010)1283 717 y Fi(0)1306 751 y Fs(\()p Fq(s)p Fs(\))p Fq(\030)t Fs(\(0\))h Fp(\000)f Fs(\010\()p Fq(s)p Fs(\))p Fq(\021)s Fs(\(0\))i Fp(\000)2073 638 y Fj(Z)2156 659 y Fl(t)2119 827 y Fk(0)2199 751 y Fs(\010\()p Fq(\033)i Fs(+)c Fq(s)p Fs(\))p Fq(g)s Fs(\()p Fq(\030)2625 763 y Fl(n)2671 751 y Fs(\()p Fq(\033)s Fs(\))p Fq(;)c(\020)2858 763 y Fl(n)2904 751 y Fs(\()p Fq(\033)s Fs(\))p Fq(;)g(\033)s Fs(\))g Fq(d\033)3244 634 y Fj(\025)661 991 y Fs(+\010\()p Fq(t)k Fs(+)g Fq(s)p Fs(\))1034 874 y Fj(\024)1078 991 y Fp(\000)p Fs(\011)1208 957 y Fi(0)1231 991 y Fs(\()p Fq(s)p Fs(\))p Fq(\030)t Fs(\(0\))h(+)f(\011\()p Fq(s)p Fs(\))p Fq(\021)s Fs(\(0\))h(+)2002 878 y Fj(Z)2085 899 y Fl(t)2048 1067 y Fk(0)2128 991 y Fs(\011\()p Fq(\033)i Fs(+)e Fq(s)p Fs(\))p Fq(g)s Fs(\()p Fq(\030)2559 1003 y Fl(n)2604 991 y Fs(\()p Fq(\033)s Fs(\))p Fq(;)14 b(\020)2791 1003 y Fl(n)2838 991 y Fs(\()p Fq(\033)s Fs(\))p Fq(;)g(\033)s Fs(\))g Fq(d\033)3178 874 y Fj(\025)495 1232 y Fq(\020)531 1244 y Fl(n)p Fk(+1)661 1232 y Fs(\()p Fq(t)p Fs(\))23 b(=)g Fq(\020)6 b Fs(\(0\))19 b(+)f(\003)p Fq(J)8 b Fs(\(0\))p Fq(t)18 b Fp(\000)1465 1119 y Fj(Z)1548 1139 y Fl(t)1511 1307 y Fk(0)1591 1232 y Fs(\003\()p Fq(t)h Fp(\000)f Fq(\033)s Fs(\))p Fq(N)9 b Fs(\()p Fq(\030)2039 1244 y Fl(n)2085 1232 y Fs(\()p Fq(\033)s Fs(\))p Fq(;)14 b(\020)2272 1244 y Fl(n)2318 1232 y Fs(\()p Fq(\033)s Fs(\))p Fq(;)g(\033)s Fs(\))p Fq(d\033)n(:)-135 1552 y Fs(F)-7 b(or)27 b(the)h(\014rst)f(iteration)g (w)n(e)g(obtain)g(\()p Fq(\030)1113 1564 y Fk(1)1151 1552 y Fq(;)14 b(\020)1224 1564 y Fk(1)1262 1552 y Fs(\))28 b(as:)650 1785 y Fq(\030)686 1797 y Fk(1)724 1785 y Fs(\()p Fq(t)p Fs(\))23 b(=)g(\011\()p Fq(t)18 b Fs(+)g Fq(s)p Fs(\))1242 1668 y Fj(\024)1286 1785 y Fs(\010)1346 1750 y Fi(0)1369 1785 y Fs(\()p Fq(s)p Fs(\))p Fq(\030)t Fs(\(0\))h Fp(\000)f Fs(\010\()p Fq(s)p Fs(\))p Fq(\021)s Fs(\(0\))i Fp(\000)2136 1672 y Fj(Z)2219 1692 y Fl(t)2182 1860 y Fk(0)2262 1785 y Fs(\010\()p Fq(\033)i Fs(+)c Fq(s)p Fs(\))p Fq(g)s Fs(\(0)p Fq(;)c Fs(0)p Fq(;)g(\033)s Fs(\))g Fq(d\033)2999 1668 y Fj(\025)816 2025 y Fs(+\010\()p Fq(t)k Fs(+)g Fq(s)p Fs(\))1189 1908 y Fj(\024)1233 2025 y Fp(\000)p Fs(\011)1363 1991 y Fi(0)1386 2025 y Fs(\()p Fq(s)p Fs(\))p Fq(\030)t Fs(\(0\))h(+)f(\011\()p Fq(s)p Fs(\))p Fq(\021)s Fs(\(0\))h(+)2157 1912 y Fj(Z)2240 1933 y Fl(t)2203 2101 y Fk(0)2283 2025 y Fs(\011\()p Fq(\033)i Fs(+)e Fq(s)p Fs(\))p Fq(g)s Fs(\(0)p Fq(;)14 b Fs(0)p Fq(;)g(\033)s Fs(\))g Fq(d\033)3025 1908 y Fj(\025)3845 2025 y Fs(\(88\))650 2265 y Fq(\020)686 2277 y Fk(1)724 2265 y Fs(\()p Fq(t)p Fs(\))23 b(=)g Fq(\020)6 b Fs(\(0\))19 b(+)f(\003)p Fq(J)8 b Fs(\(0\))p Fq(t)18 b Fp(\000)1528 2152 y Fj(Z)1611 2173 y Fl(t)1574 2341 y Fk(0)1654 2265 y Fs(\003\()p Fq(t)h Fp(\000)f Fq(\033)s Fs(\))p Fq(N)9 b Fs(\(0)p Fq(;)14 b Fs(0)p Fq(;)g(\033)s Fs(\))p Fq(d\033)1449 b Fs(\(89\))-260 2487 y(This)28 b(\014rst)f(iterate)g(can)g(easily)g(b) r(e)h(b)r(ounded,)g(using)f(the)h(initial)g(conditions)f(\(86\),)h(and) f(Lemmas)g(14)g(and)g(15:)634 2711 y Fp(j)p Fq(\030)693 2723 y Fk(1)731 2711 y Fs(\()p Fq(t)p Fs(\))p Fp(j)c(\026)969 2655 y Fq(\026)p 969 2692 51 4 v 971 2768 a(\034)1043 2594 y Fj(\024)1131 2655 y Fs(1)p 1097 2692 111 4 v 1097 2768 a Fq(\016)1137 2744 y Fl(p)1171 2752 y Fe(3)1236 2711 y Fs(+)1319 2619 y Fj(\020)1369 2711 y Fq(\032)1412 2668 y Fi(\000)p Fk(\(2)p Fl(l)p Fi(\000)p Fk(3\))1412 2739 y([)p Fl(t;)p Fk(0])1659 2711 y Fs(\()p Fq(s)p Fs(\))c(+)f(1)1906 2619 y Fj(\021)2002 2655 y Fs(1)p 1979 2692 88 4 v 1979 2768 a Fq(\016)2019 2744 y Fl(\013)2076 2594 y Fj(\025)2138 2711 y Fs(+)g Fq(\026\034)2316 2677 y Fk(2)2368 2594 y Fj(\024)2456 2655 y Fs(1)p 2422 2692 111 4 v 2422 2768 a Fq(\016)2462 2744 y Fl(p)2496 2752 y Fe(3)2561 2711 y Fs(+)g Fq(\032)2687 2675 y Fi(\000)p Fk(2)p Fl(l)2687 2739 y Fk([)p Fl(t;)p Fk(0])2807 2711 y Fs(\()p Fq(s)p Fs(\))2943 2655 y(1)p 2920 2692 88 4 v 2920 2768 a Fq(\016)2960 2744 y Fl(\013)3017 2594 y Fj(\025)3075 2711 y Fq(:)747 b Fs(\(90\))-260 2944 y(An)28 b(analogous)e(b)r(ound)h(for)h Fq(\034)9 b(\030)724 2956 y Fk(1)762 2944 y Fs(\()p Fq(t)p Fs(\))28 b(follo)n(ws)f(immediately)-7 b(.)37 b(No)n(w,)27 b(using)g(that)h(for)f Fq(s)c Fp(2)h Fq(D)2603 2914 y Fk(do)n(wn)2786 2944 y Fs(w)n(e)k(ha)n(v)n(e)1600 3118 y Fq(\032)1643 3083 y Fi(\000)p Fk(2)p Fl(l)1643 3147 y Fk([)p Fl(t;)p Fk(0])1762 3118 y Fs(\()p Fq(s)p Fs(\))c Fp(\026)e Fq(\034)2021 3084 y Fi(\000)p Fk(2)p Fl(l)-260 3302 y Fs(and)28 b(for)f Fq(s)c Fp(2)g Fq(D)240 3272 y Fk(up)345 3302 y Fs(w)n(e)k(ha)n(v)n(e)1120 3477 y Fq(\032)1163 3441 y Fi(\000)p Fk(2)p Fl(l)1163 3505 y Fk([)p Fl(t;)p Fk(0])1283 3477 y Fs(\()p Fq(s)p Fs(\))c Fp(\026)g Fq(\034)1542 3442 y Fi(\000)p Fk(2)p Fl(l)1653 3477 y Fq(;)83 b Fs(for)g(0)22 b Fp(\024)h Fq(t)18 b Fs(+)g(Re)c Fq(s)23 b Fp(\024)g Fs(Re)14 b Fq(s;)735 3629 y(\032)778 3593 y Fi(\000)p Fk(2)p Fl(l)778 3657 y Fk([)p Fl(t;)p Fk(0])898 3629 y Fs(\()p Fq(s)p Fs(\))23 b Fp(\026)g Fq(\016)1152 3594 y Fi(\000)p Fk(2)p Fl(l)1262 3629 y Fq(;)97 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Fs(Assuming)27 b(that)h Fp(k)p Fs(\()p Fq(\030)536 -70 y Fl(n)p Fi(\000)p Fk(1)666 -82 y Fq(;)14 b(\020)739 -70 y Fl(n)p Fi(\000)p Fk(1)870 -82 y Fs(\))p Fp(k)f Fq(;)h Fp(k)p Fs(\()p Fq(\030)1104 -70 y Fl(n)1150 -82 y Fq(;)g(\020)1223 -70 y Fl(n)1268 -82 y Fs(\))p Fp(k)23 b(\026)f Fq(\026)1516 -149 y Fj(\000)1598 -115 y Fk(1)p 1564 -101 100 4 v 1564 -54 a Fl(\016)1596 -72 y Ff(p)1627 -60 y Fe(3)1692 -82 y Fs(+)1888 -115 y Fk(1)p 1785 -101 239 4 v 1785 -53 a Fl(\016)1817 -69 y Fe(2)p Ff(l)p Fe(+)p Ff(\013)p Fh(\000)p Fe(2)2034 -149 y Fj(\001)2072 -82 y Fs(,)27 b(w)n(e)h(no)n(w)f(consider)783 145 y Fq(\030)819 157 y Fl(n)p Fk(+1)948 145 y Fs(\()p Fq(t)p Fs(\))19 b Fp(\000)g Fq(\030)1181 157 y Fl(n)1226 145 y Fs(\()p Fq(t)p Fs(\))84 b(=)101 b Fp(\000)18 b Fs(\011\()p Fq(t)g Fs(+)g Fq(s)p Fs(\))1966 32 y Fj(Z)2049 53 y Fl(t)2012 221 y Fk(0)2092 145 y Fs(\010\()p Fq(\033)k Fs(+)c Fq(s)p Fs(\))c([)p Fq(g)2484 157 y Fl(n)2548 145 y Fp(\000)k Fq(g)2671 157 y Fl(n)p Fi(\000)p Fk(1)2800 145 y Fs(])c Fq(d\033)1570 382 y Fs(+)k(\010\()p Fq(t)g Fs(+)g Fq(s)p Fs(\))1961 269 y Fj(Z)2044 290 y Fl(t)2008 458 y Fk(0)2087 382 y Fs(\011\()p Fq(\033)k Fs(+)c Fq(s)p Fs(\))c([)p Fq(g)2484 394 y Fl(n)2548 382 y Fp(\000)k Fq(g)2671 394 y Fl(n)p Fi(\000)p Fk(1)2800 382 y Fs(])c Fq(d\033)n(;)-260 598 y Fs(where)33 b Fq(g)26 610 y Fl(n)104 598 y Fs(denotes)h Fq(g)s Fs(\()p Fq(\030)522 610 y Fl(n)567 598 y Fs(\()p Fq(\033)s Fs(\))p Fq(;)14 b(\020)754 610 y Fl(n)800 598 y Fs(\()p Fq(\033)s Fs(\))p Fq(;)g(\033)s Fs(\).)57 b(No)n(w,)34 b(w)n(e)f(apply)g(part)g(\(d\))i(of)e(Lemma)g(14,)h(with)g Fq(\025)g Fs(=)e Fq(\026)p Fs(\()3067 565 y Fk(1)p 3034 579 100 4 v 3034 626 a Fl(\016)3066 608 y Ff(p)3097 620 y Fe(3)3166 598 y Fs(+)3366 565 y Fk(1)p 3263 579 239 4 v 3263 627 a Fl(\016)3295 611 y Fe(2)p Ff(l)p Fe(+)p Ff(\013)p Fh(\000)p Fe(2)3511 598 y Fs(\),)k Fq(\014)h Fs(=)32 b(1)h(and)-260 697 y(\012)23 b(=)-14 660 y Fl(\026)p -79 678 171 4 v -79 728 a(\016)-47 709 y Ff(p)-16 721 y Fe(3)16 709 y(+2)115 697 y Fs(+)266 660 y Fl(\026)p 203 678 166 4 v 203 727 a(\016)235 710 y Fe(2)p Ff(l)p Fe(+)p Ff(\013)379 697 y Fs(.)36 b(By)25 b(the)h(h)n(yp)r(otheses)e(of) h(Theorem)g(13,)g(these)g(amoun)n(ts)g(v)n(erify)f(that)i Fq(\025)d Fp(\026)g Fq(\016)2848 667 y Fk(3)2908 697 y Fp(\026)g Fs(1,)j(and)f(\012)e Fp(\026)f Fq(\016)s Fs(,)k(whic)n(h)g(are)e(the)-260 797 y(conditions)29 b(needed)h(in)h(this)f(lemma.)43 b(W)-7 b(e)30 b(will)g(use)g(the)g (notation)f Fq(\034)9 b Fs(\()p Fq(\033)s Fs(\))29 b(=)d Fp(j)p Fq(\033)c Fs(+)c Fq(T)30 b Fp(\000)18 b Fq(\031)s(i=)p Fs(2)p Fp(j)o Fs(,)30 b(and)g(w)n(e)f(will)h(use)g(again)e(Lemma)i(15,) -260 897 y(as)d(w)n(ell)g(as)g(inequalit)n(y)h(\(91\).)36 b(W)-7 b(e)28 b(obtain:)632 1058 y Fp(j)p Fq(\034)9 b Fs(\()p Fq(\030)768 1070 y Fl(n)p Fk(+1)899 1058 y Fs(\()p Fq(t)p Fs(\))19 b Fp(\000)f Fq(\030)1131 1070 y Fl(n)1176 1058 y Fs(\()p Fq(t)p Fs(\)\))p Fp(j)715 1240 y(\026)863 1127 y Fj(Z)946 1147 y Fl(t)909 1315 y Fk(0)989 1240 y Fq(\034)9 b Fs(\()p Fq(\033)s Fs(\))1148 1206 y Fk(2)1201 1123 y Fj(\024)o(\022)1390 1184 y Fq(\025)p 1315 1221 198 4 v 1315 1297 a(\034)g Fs(\()p Fq(\033)s Fs(\))1474 1273 y Fk(3)1542 1240 y Fs(+)1650 1184 y Fq(\026\016)1740 1154 y Fi(\000)p Fl(\013)p 1635 1221 220 4 v 1635 1297 a Fq(\034)g Fs(\()p Fq(\033)s Fs(\))1794 1273 y Fk(2)p Fl(l)1864 1123 y Fj(\023)1939 1240 y Fp(j)p Fq(\030)1998 1252 y Fl(n)2043 1240 y Fs(\()p Fq(\033)s Fs(\))20 b Fp(\000)e Fq(\030)2296 1252 y Fl(n)p Fi(\000)p Fk(1)2426 1240 y Fs(\()p Fq(\033)s Fs(\))p Fp(j)1228 1482 y Fs(+)1331 1426 y Fq(\026\016)1421 1396 y Fi(\000)p Fk(\()p Fl(\013)p Fk(+1\))p 1331 1463 326 4 v 1384 1539 a Fq(\034)9 b Fs(\()p Fq(\033)s Fs(\))1543 1515 y Fk(2)p Fl(l)1680 1482 y Fp(j)p Fq(\020)1739 1494 y Fl(n)1785 1482 y Fs(\()p Fq(\033)s Fs(\))19 b Fp(\000)f Fq(\020)2037 1494 y Fl(n)p Fi(\000)p Fk(1)2168 1482 y Fs(\()p Fq(\033)s Fs(\))p Fp(j)2306 1365 y Fj(\025)2363 1482 y Fq(d\033)881 1722 y Fs(+)g Fq(\034)1009 1688 y Fk(3)1061 1609 y Fj(Z)1144 1630 y Fl(t)1107 1798 y Fk(0)1257 1666 y Fs(1)p 1197 1703 161 4 v 1197 1779 a Fq(\034)9 b Fs(\()p Fq(\033)s Fs(\))1381 1605 y Fj(\024\022)1571 1666 y Fq(\025)p 1496 1703 198 4 v 1496 1779 a(\034)g Fs(\()p Fq(\033)s Fs(\))1655 1755 y Fk(3)1723 1722 y Fs(+)1831 1666 y Fq(\026\016)1921 1636 y Fi(\000)p Fl(\013)p 1816 1703 220 4 v 1816 1779 a Fq(\034)g Fs(\()p Fq(\033)s Fs(\))1975 1755 y Fk(2)p Fl(l)2045 1605 y Fj(\023)2120 1722 y Fp(j)p Fq(\030)2179 1734 y Fl(n)2224 1722 y Fs(\()p Fq(\033)s Fs(\))20 b Fp(\000)e Fq(\030)2477 1734 y Fl(n)p Fi(\000)p Fk(1)2607 1722 y Fs(\()p Fq(\033)s Fs(\))p Fp(j)1409 1964 y Fs(+)1512 1908 y Fq(\026\016)1602 1878 y Fi(\000)p Fk(\()p Fl(\013)p Fk(+1\))p 1512 1945 326 4 v 1565 2021 a Fq(\034)9 b Fs(\()p Fq(\033)s Fs(\))1724 1997 y Fk(2)p Fl(l)1861 1964 y Fp(j)p Fq(\020)1920 1976 y Fl(n)1966 1964 y Fs(\()p Fq(\033)s Fs(\))19 b Fp(\000)f Fq(\020)2218 1976 y Fl(n)p Fi(\000)p Fk(1)2349 1964 y Fs(\()p Fq(\033)s Fs(\))p Fp(j)2487 1847 y Fj(\025)2544 1964 y Fq(d\033)715 2172 y Fp(\026)863 2079 y Fj(h\020)951 2172 y Fq(\025\032)1042 2136 y Fi(\000)p Fk(1)1042 2200 y([)p Fl(t;)p Fk(0])1162 2172 y Fs(\()p Fq(s)p Fs(\))h(+)f Fq(\026\016)1457 2137 y Fi(\000)p Fl(\013)1557 2172 y Fq(\032)1600 2128 y Fi(\000)p Fk(\(2)p Fl(l)p Fi(\000)p Fk(2\))1600 2200 y([)p Fl(t;)p Fk(0])1847 2172 y Fs(\()p Fq(s)p Fs(\))h(+)f Fq(\026\016)2142 2137 y Fi(\000)p Fk(\()p Fl(\013)p Fk(+3\))2391 2079 y Fj(\020)2441 2172 y Fs(1)f(+)i Fq(\032)2627 2128 y Fi(\000)p Fk(\(2)p Fl(l)p Fi(\000)p Fk(3\))2627 2200 y([)p Fl(t;)p Fk(0])2874 2172 y Fs(\()p Fq(s)p Fs(\))2977 2079 y Fj(\021\021)910 2354 y Fs(+)p Fq(\034)1020 2320 y Fk(3)1071 2262 y Fj(\020)1120 2354 y Fq(\025\032)1211 2319 y Fi(\000)p Fk(4)1211 2383 y([)p Fl(t;)p Fk(0])1331 2354 y Fs(\()p Fq(s)p Fs(\))g(+)f Fq(\026\016)1626 2320 y Fi(\000)p Fl(\013)1726 2354 y Fq(\032)1769 2311 y Fi(\000)p Fk(\(2)p Fl(l)p Fk(+1\))1769 2383 y([)p Fl(t;)p Fk(0])2015 2354 y Fs(\()p Fq(s)p Fs(\))h(+)f Fq(\026\016)2310 2320 y Fi(\000)p Fk(\()p 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5715 y Fk(1)1383 5703 y Fs(\()p Fq(t)p Fs(\))p Fp(j)d(\026)e Fq(\026)1675 5586 y Fj(\022)1823 5647 y Fs(1)p 1746 5684 195 4 v 1746 5760 a Fq(\016)1786 5736 y Fl(p)1820 5744 y Fe(3)1853 5736 y Fk(+2)1969 5703 y Fs(+)2138 5647 y(1)p 2062 5684 193 4 v 2062 5760 a Fq(\016)2102 5736 y Fk(2)p Fl(l)p Fk(+)p Fl(\013)2265 5586 y Fj(\023)2340 5703 y Fq(\034)2385 5669 y Fk(2)2423 5703 y Fq(:)1825 5989 y Fs(30)p eop %%Page: 31 31 31 30 bop -260 -82 a Fs(F)-7 b(or)27 b(the)i(successiv)n(e)d(iterates)h (w)n(e)h(apply)g(Lemma)f(14,)g(with)i Fq(\025)24 b Fs(=)f(\012)g(=)h Fq(\026)2081 -149 y Fj(\000)2197 -115 y Fk(1)p 2128 -101 171 4 v 2128 -52 a Fl(\016)2160 -70 y Ff(p)2191 -58 y Fe(3)2224 -70 y(+2)2327 -82 y Fs(+)2486 -115 y Fk(1)p 2420 -101 166 4 v 2420 -53 a Fl(\016)2452 -69 y Fe(2)p Ff(l)p Fe(+)p Ff(\013)2595 -149 y Fj(\001)2657 -82 y Fp(\026)f Fq(\016)s Fs(,)28 b Fq(\014)g Fs(=)23 b Fp(\000)p Fs(2,)28 b(and)f(Lemma)h(15,)f(as)g(w)n(ell)-260 17 y(as)g (inequalities)g(\(93\),)h(obtaining)565 135 y 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1705 y Fs(+)g Fq(\026\016)1818 1671 y Fi(\000)p Fk(\()p Fl(\013)p Fk(+1\))2053 1705 y Fq(\032)2096 1671 y Fi(\000)p Fk(2)p Fl(l)2206 1613 y Fj(\021)2270 1705 y Fp(k)o Fs(\()p Fq(\030)2379 1717 y Fl(n)2425 1705 y Fq(;)c(\020)2498 1717 y Fl(n)2543 1705 y Fs(\))19 b Fp(\000)f Fs(\()p Fq(\030)2745 1717 y Fl(n)p Fi(\000)p Fk(1)2876 1705 y Fq(;)c(\020)2949 1717 y Fl(n)p Fi(\000)p Fk(1)3079 1705 y Fs(\))p Fp(k)648 1888 y(\026)796 1796 y Fj(\020)928 1832 y Fq(\026)p 855 1869 195 4 v 855 1945 a(\016)895 1921 y Fl(p)929 1929 y Fe(3)962 1921 y Fk(+2)1079 1888 y Fs(+)1285 1832 y Fq(\026)p 1172 1869 277 4 v 1172 1945 a(\016)1212 1921 y Fk(2)p Fl(l)p Fk(+)p Fl(\013)p Fk(+1)1459 1796 y Fj(\021)1522 1888 y Fp(k)o Fs(\()p Fq(\030)1631 1900 y Fl(n)1677 1888 y Fq(;)g(\020)1750 1900 y Fl(n)1796 1888 y Fs(\))k Fp(\000)g Fs(\()p Fq(\030)1997 1900 y Fl(n)p Fi(\000)p Fk(1)2128 1888 y Fq(;)c(\020)2201 1900 y Fl(n)p Fi(\000)p Fk(1)2332 1888 y Fs(\))p Fp(k)f Fq(;)-260 2122 y Fs(and)33 b(similarly)f(for)g Fp(j)q 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4809 y Fq(q)1002 4821 y Fk(2)1062 4809 y Fp(\025)23 b Fs(Re)17 b(~)-45 b Fq(s)23 b Fp(\025)f(\000)p Fq(q)1513 4821 y Fi(\003)1551 4809 y Fq(;)180 b Fp(j)p Fs(Im)17 b(~)-45 b Fq(s)p Fp(j)23 b(\024)g Fq(\027)2104 4821 y Fk(1)2141 4809 y Fq(;)2344 4714 y Fj(\014)2344 4764 y(\014)2344 4814 y(\014)2372 4809 y Fs(Im)2491 4787 y(~)2485 4809 y Fq(\022)2526 4714 y Fj(\014)2526 4764 y(\014)2526 4814 y(\014)2577 4809 y Fp(\024)f Fq(\032)2707 4821 y Fk(3)2744 4809 y Fq(:)-260 5031 y Fs(T)-7 b(o)34 b(apply)g(Theorem)f(13,)i (consider)e(the)i(initial)f(parameter)f Fq(s)h Fs(=)f(2)p Fq(q)1983 5043 y Fk(2)2043 5031 y Fs(+)22 b Fq(i)14 b Fs(Im)j(~)-45 b Fq(s)34 b Fs(and)g(time)g Fq(t)g Fs(=)2895 4998 y Fk(1)p 2895 5012 34 4 v 2897 5059 a Fl(b)2938 5031 y Fs(\(Re)17 b(~)-45 b Fq(s)23 b Fp(\000)f Fs(2)p Fq(q)3310 5043 y Fk(2)3347 5031 y Fs(\))34 b Fp(\024)g Fs(0.)56 b(Note)34 b(that)-260 5145 y Fq(s)18 b Fs(+)g Fq(bt)23 b Fs(=)j(~)-46 b Fq(s)p Fs(,)28 b(and)f(consider)g(also)f Fq(\022)k 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Fl(")2251 5458 y Fq(t)p Fs(\))19 b Fp(\000)f Fq(Z)2472 5470 y Fk(0)2509 5458 y Fs(\()p Fq(s)h Fs(+)f Fq(bt;)c(\022)20 b Fs(+)k(~)-48 b Fq(!)2979 5470 y Fl(")3014 5458 y Fq(t)p Fs(\))3076 5387 y Fj(\014)3076 5437 y(\014)841 5616 y Fp(\024)989 5545 y Fj(\014)989 5595 y(\014)1016 5616 y Fq(Z)1079 5581 y Fk(+)1134 5616 y Fs(\()p Fq(s)19 b Fs(+)f Fq(bt;)c(\022)20 b Fs(+)25 b(~)-49 b Fq(!)1604 5628 y Fl(")1639 5616 y Fq(t)p Fs(\))19 b Fp(\000)f Fq(Z)1860 5628 y Fk(0)1897 5616 y Fs(\()p Fq(s)h Fs(+)f Fq(t;)c(\022)20 b Fs(+)e Fq(!)2331 5628 y Fl(")2366 5616 y Fq(t)p Fs(\))2428 5545 y Fj(\014)2428 5595 y(\014)989 5740 y Fs(+)c Fp(j)o Fq(Z)1147 5752 y Fk(0)1184 5740 y Fs(\()p Fq(s)19 b Fs(+)f Fq(t;)c(\022)20 b Fs(+)e Fq(!)1618 5752 y Fl(")1654 5740 y Fq(t)p Fs(\))g Fp(\000)g Fq(Z)1874 5752 y Fk(0)1911 5740 y Fs(\()p Fq(s)h Fs(+)f Fq(bt;)c(\022)20 b Fs(+)25 b(~)-49 b Fq(!)2381 5752 y Fl(")2416 5740 y Fq(t)p Fs(\))p Fp(j)1825 5989 y Fs(31)p eop %%Page: 32 32 32 31 bop 841 -45 a Fp(\026)989 -137 y Fj(\020)1079 -101 y Fq(\026)p 1048 -64 111 4 v 1048 12 a(\016)1088 -12 y Fl(p)1122 -4 y Fe(5)1169 -45 y Fq(;)1246 -101 y(\026)p 1216 -64 V 1216 12 a(\016)1256 -12 y Fl(p)1290 -4 y Fe(4)1337 -137 y Fj(\021)1405 -45 y Fs(+)1488 -162 y Fj(\022)1559 -101 y Fp(j)p Fq(b)18 b Fp(\000)g Fs(1)p Fp(j)p 1559 -64 225 4 v 1633 12 a Fq(\016)1673 -12 y Fk(2)1794 -45 y Fq(;)c Fp(j)p Fq(!)1906 -33 y Fl(")1959 -45 y Fp(\000)25 b Fs(~)-49 b Fq(!)2094 -33 y Fl(")2129 -45 y Fp(j)2152 -162 y Fj(\023)841 188 y Fp(\026)989 71 y Fj(\022)1090 131 y Fq(\026)p 1060 168 111 4 v 1060 245 a(\016)1100 221 y Fl(p)1134 229 y Fe(5)1199 188 y Fs(+)1365 131 y Fq(\026)p 1292 168 195 4 v 1292 245 a(\016)1332 221 y Fl(p)1366 229 y Fe(2)1399 221 y Fk(+2)1497 188 y Fq(;)1574 131 y(\026)p 1544 168 111 4 v 1544 245 a(\016)1584 221 y Fl(p)1618 229 y Fe(4)1683 188 y Fs(+)1860 131 y Fq(\026)p 1776 168 219 4 v 1776 245 a(\016)1816 221 y Fl(p)1850 229 y Fe(2)1887 185 y Fp(p)p 1956 185 39 4 v 60 x Fq(")2005 71 y Fj(\023)2089 188 y Fp(\026)2176 71 y Fj(\022)2278 131 y Fq(\026)p 2248 168 111 4 v 2248 245 a(\016)2288 221 y Fl(p)2322 229 y Fe(5)2368 188 y Fq(;)2446 131 y(\026)p 2415 168 V 2415 245 a(\016)2455 221 y Fl(p)2489 229 y Fe(4)2554 188 y Fs(+)2732 131 y Fq(\026)p 2647 168 219 4 v 2647 245 a(\016)2687 221 y Fl(p)2721 229 y Fe(2)2758 185 y Fp(p)p 2827 185 39 4 v 60 x Fq(")2876 71 y Fj(\023)2951 188 y Fq(:)871 b Fs(\(94\))-135 512 y(In)21 b(order)f(to)i(b)r(ound)f (the)h(in)n(tegral)e(in)i(statemen)n(t)f(\(b\))i(of)e(Theorem)g(6,)h(w) n(e)f(use)g(the)h(previous)e(results)h(and)g(part)g(\(d\))i(of)e(Lemma) g(14.)-260 612 y(T)-7 b(o)27 b(this)h(end)g(w)n(e)f(de\014ne)486 787 y Fq(\030)522 799 y Fk(1)642 787 y Fs(=)83 b Fq(Z)853 753 y Fk(+)847 807 y Fl(x)907 787 y Fs(\()p Fq(s)19 b Fs(+)f Fq(bt;)c(\022)20 b Fs(+)25 b(~)-49 b Fq(!)1377 799 y Fl(")1413 787 y Fq(t)p Fs(\))18 b Fp(\000)g Fq(x)1623 799 y Fk(0)1661 787 y Fs(\()p Fq(s)h Fs(+)f Fq(t)p Fs(\))p Fq(;)97 b(\020)2052 799 y Fk(1)2113 787 y Fs(=)22 b Fq(Z)2263 753 y Fk(+)2257 807 y Fl(')2318 787 y Fs(\()p Fq(s)d Fs(+)f Fq(bt;)c(\022)20 b Fs(+)k(~)-48 b Fq(!)2788 799 y Fl(")2823 787 y Fq(t)p Fs(\))19 b Fp(\000)f Fq(\022)j Fp(\000)d Fq(!)3182 799 y Fl(")3217 787 y Fq(t)486 911 y(\030)522 923 y Fk(2)642 911 y Fs(=)83 b Fq(x)837 923 y Fk(0)874 911 y Fs(\()p Fq(s)19 b Fs(+)f Fq(bt)p Fs(\))g Fp(\000)g Fq(x)1293 923 y Fk(0)1331 911 y Fs(\()p Fq(s)h Fs(+)f Fq(t)p Fs(\))p Fq(;)97 b(\020)1722 923 y Fk(2)1783 911 y Fs(=)22 b Fq(\022)f Fs(+)k(~)-49 b Fq(!)2065 923 y Fl(")2100 911 y Fq(t)19 b Fp(\000)f Fq(\022)i Fp(\000)e Fq(!)2426 923 y Fl(")2462 911 y Fq(t:)-260 1087 y Fs(By)27 b(the)h(results)f(of)h(Theorem)f(13)g(w)n(e)g(ha)n(v)n(e)1294 1203 y Fp(j)p Fq(\030)1353 1215 y Fk(1)1390 1203 y Fp(j)c(\026)1564 1147 y Fq(\026)p 1534 1184 111 4 v 1534 1260 a(\016)1574 1236 y Fl(p)1608 1244 y Fe(5)1678 1203 y Fp(\026)f Fs(1)p Fq(;)97 b Fp(j)p Fq(\020)1986 1215 y Fk(1)2023 1203 y Fp(j)24 b(\026)2198 1147 y Fq(\026)p 2167 1184 V 2167 1260 a(\016)2207 1236 y Fl(p)2241 1244 y Fe(4)2311 1203 y Fp(\026)f Fq(\016)-260 1371 y Fs(and,)28 b(b)n(y)f(Theorem)g(1)g(w)n (e)g(ha)n(v)n(e)879 1590 y Fp(j)p Fq(\030)938 1602 y Fk(2)976 1590 y Fp(j)c(\026)1119 1533 y(j)p Fq(b)18 b Fp(\000)g Fs(1)p Fp(j)p 1119 1570 225 4 v 1212 1646 a Fq(\016)1377 1590 y Fp(\026)1547 1533 y Fq(\026)p 1475 1570 195 4 v 1475 1646 a(\016)1515 1622 y Fl(p)1549 1630 y Fe(2)1581 1622 y Fk(+1)1703 1590 y Fp(\026)k Fs(1)p Fq(;)97 b Fp(j)p Fq(\020)2011 1602 y Fk(2)2048 1590 y Fp(j)23 b(\026)g(j)7 b Fs(~)-49 b Fq(!)2257 1602 y Fl(")2311 1590 y Fp(\000)18 b Fq(!)2446 1602 y Fl(")2481 1590 y Fp(j)23 b(\026)2709 1533 y Fq(\026)p 2625 1570 219 4 v 2625 1647 a(\016)2665 1623 y Fl(p)2699 1631 y Fe(2)2735 1587 y Fp(p)p 2805 1587 39 4 v 2805 1647 a Fq(")-260 1813 y Fs(and)30 b(the)h(h)n(yp)r(otheses)e(of)h(Theorem)f(6)h(imply)h Fp(j)p Fq(\020)1292 1825 y Fk(2)1329 1813 y Fp(j)d(\026)e Fq(\016)s Fs(,)31 b(and)f(hence)h(w)n(e)e(can)h(apply)g(part)g(\(d\))g (of)h(Lemma)e(14.)44 b(Note)30 b(also)f(that)i(w)n(e)-260 1912 y(ha)n(v)n(e)g(giv)n(en)g(b)r(ounds)h(for)g Fp(j)p Fq(\030)641 1924 y Fk(1)697 1912 y Fp(\000)18 b Fq(\030)816 1924 y Fk(2)854 1912 y Fp(j)32 b Fs(and)g Fp(j)p Fq(\020)1134 1924 y Fk(1)1190 1912 y Fp(\000)18 b Fq(\020)1309 1924 y Fk(2)1346 1912 y Fp(j)32 b Fs(in)h(\(94\).)50 b(Applying)32 b(also)f(Lemma)g(15,)i(w)n(e)e(obtain)h(the)h(desired)e(b)r(ound)h (\(b\))h(of)-260 2012 y(Theorem)27 b(6:)521 2096 y Fj(\014)521 2146 y(\014)521 2195 y(\014)521 2245 y(\014)521 2295 y(\014)549 2128 y(Z)632 2149 y Fl(T)595 2317 y Fk(0)698 2174 y Fj(\002)732 2241 y Fq(@)776 2253 y Fl(')824 2241 y Fq(H)893 2253 y Fk(1)931 2241 y Fs(\()p Fq(Z)1026 2207 y Fk(+)1081 2241 y Fs(\()p Fq(s)18 b Fs(+)g Fq(bt;)c(\022)20 b Fs(+)25 b(~)-49 b Fq(!)1550 2253 y Fl(")1586 2241 y Fq(t)p Fs(\)\))19 b Fp(\000)f Fq(@)1826 2253 y Fl(')1874 2241 y Fq(H)1943 2253 y Fk(1)1980 2241 y Fs(\()p Fq(Z)2069 2253 y Fk(0)2106 2241 y Fs(\()p Fq(s)h Fs(+)f Fq(bt;)c(\022)20 b Fs(+)25 b(~)-49 b Fq(!)2576 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y Fs(+)15 b Fq(R)q Fo(,)29 b(analytic)h(on)e Fp(B)2113 1606 y Fl(\024;\033)o(;\021)r(;\032;\020)2376 1594 y Fo(,)h(with)g Fq(K)2680 1606 y Fk(0)2740 1594 y Fs(=)23 b Fp(h)7 b Fs(~)-49 b Fq(!)2912 1606 y Fl(")2947 1594 y Fq(;)14 b(I)7 b Fp(i)16 b Fs(+)f Fq(bE)28 b Fs(=)23 b Fp(h)p Fq(b)7 b Fs(^)-49 b Fq(!)3488 1606 y Fl(")3523 1594 y Fq(;)14 b(I)7 b Fp(i)15 b Fs(+)h Fq(bE)33 b Fo(and)-260 1705 y Fq(R)24 b Fs(=)f Fp(O)-19 1717 y Fk(2)18 1705 y Fs(\()p Fq(E)5 b(;)14 b(I)7 b Fs(\))p Fo(.)39 b(F)-6 b(or)33 b Fs(^)-45 b Fq(\024)23 b(<)g(\024)p Fo(,)34 b Fs(^)-46 b Fq(\033)26 b(<)d(\033)s Fo(,)36 b Fs(^)-48 b Fq(\021)27 b(<)22 b(\021)s Fo(,)38 b Fs(^)-49 b Fq(\032)23 b(<)f(\032)p Fo(,)1489 1683 y Fs(^)1481 1705 y Fq(\020)30 b(<)22 b(\020)37 b Fo(given,)30 b(assume)g(that)1266 2002 y Fp(k)p Fq(R)q Fp(k)1413 2027 y Fl(\024;\033)o(;\021)r (;\032;\020)1700 2002 y Fp(\024)1797 1915 y Fs(min)1949 1823 y Fj(\020)2002 1915 y Fs(^)-45 b Fq(\024)6 b Fs(^)-48 b Fq(\021)s(;)18 b Fs(^)-46 b Fq(\033)9 b Fs(^)-48 b Fq(\021)t(;)14 b(e)8 b Fs(^)-50 b Fq(\032)2348 1894 y Fs(^)2341 1915 y Fq(\020)2383 1823 y Fj(\021)p 1797 1983 636 4 v 2038 2059 a Fs(8)p Fq(e)2119 2035 y Fk(2)2156 2059 y Fq(c)2443 2002 y(;)1379 b Fs(\(95\))-260 2215 y Fo(wher)l(e)31 b Fq(c)23 b Fs(=)f(2\()p Fq(\024)c Fs(+)g(2)p Fq(\033)s Fs(\))p Fo(.)39 b(Then,)31 b(ther)l(e)f(exists)f(a)h(symple)l (ctic)h(map)1123 2400 y Fs(\007)1188 2366 y Fk(\(1\))1300 2400 y Fs(:)23 b Fp(B)1401 2427 y Fl(\024)p Fi(\000)s Fk(^)-36 b Fl(\024)o(;\033)r Fi(\000)t Fk(^)f Fl(\033)s(;\021)r Fi(\000)5 b Fk(^)-38 b Fl(\021)r(;\032)p Fi(\000)6 b Fk(^)-39 b Fl(\032)q(;\020)t Fi(\000)2080 2412 y Fk(^)2074 2427 y Fl(\020)2134 2400 y Fp(\000)-14 b(!)23 b(B)2346 2412 y Fl(\024;\033)o(;\021)r(;\032;\020)-260 2586 y Fo(such)30 b(that)1464 2689 y Fq(K)24 b Fp(\016)18 b Fs(\007)1684 2655 y Fk(\()o(1\))1796 2689 y Fs(=)k Fq(K)1954 2701 y Fk(0)2009 2689 y Fs(+)c Fq(R)2156 2655 y Fk(\(1\))2245 2689 y Fq(;)-260 2843 y Fo(with)30 b Fs(\007)-15 2812 y 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y(W)-7 b(e)19 b(\014rst)g (establish)f(the)i(follo)n(wing)e(lemma,)i(whic)n(h)f(pro)n(vides)f(an) g(estimate)h(for)g(a)f(solution)h(of)g(the)g(linearized)f(equation)g (app)r(earing)-260 3836 y(in)28 b(the)g(pro)r(of)g(of)f(Lemma)h(18.)37 b(W)-7 b(e)28 b(p)r(oin)n(t)g(out)g(that)g(suc)n(h)f(an)h(estimate)g (do)r(es)f(not)h(in)n(v)n(olv)n(e)e(small)i(divisors.)36 b(Its)28 b(only)f(di\016cult)n(y)h(lies)-260 3936 y(in)d(the)g(c)n (hoice)f(of)h(the)g(suitable)g(solution)f(to)h(b)r(e)g(estimated.)36 b(Our)24 b(approac)n(h)f(is)h(analogous)f(to)h(the)i(one)e(in)n(tro)r (duced)h(in)g([Sau01)n(])g(\(see)-260 4036 y(also)i([R)-9 b(W00)o(]\),)28 b(and)f(w)n(e)h(giv)n(e)e(here)h(the)h(pro)r(of)f(for)h (completeness.)-260 4322 y Ft(Lemma)i(19)41 b Fo(L)l(et)29 b Fq(R)h Fo(analytic)h(on)f Fp(B)940 4334 y Fl(\024;\033)o(;\021)r (;\032;\020)1203 4322 y Fo(.)39 b(Then,)31 b(the)f(line)l(ar)g(e)l (quation)1445 4549 y Fq(@)1489 4561 y Fl(S)1538 4549 y Fq(U)d Fs(+)18 b Fp(h)7 b Fs(^)-49 b Fq(!)1789 4561 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(eac)n(h)f(co)r(e\016cien)n(t)g Fq(U)1392 5413 y Fl(k)1459 5401 y Fs(has)g(to)g(b)r(e)h(a)f(solution)g(of)g(the)h(follo)n(wing)f (simple)g(linear)g(ordinary)e(di\013eren)n(tial)-260 5500 y(equation)j(\(with)i(resp)r(ect)e(to)g(the)h(v)-5 b(ariable)27 b Fq(S)5 b Fs(\):)1377 5725 y Fq(@)1421 5737 y Fl(S)1469 5725 y Fq(U)1526 5737 y Fl(k)1585 5725 y Fs(+)18 b Fq(i)c Fp(h)p Fq(k)s(;)20 b Fs(^)-48 b Fq(!)1878 5737 y Fl(")1913 5725 y Fp(i)14 b Fq(U)2016 5737 y Fl(k)2079 5725 y Fs(=)2177 5669 y(1)p 2177 5706 V 2180 5782 a Fq(b)2228 5725 y(R)2291 5737 y Fl(k)2332 5725 y Fq(:)1825 5989 y Fs(33)p eop %%Page: 34 34 34 33 bop -260 -82 a Fs(F)-7 b(ollo)n(wing)27 b([Sau01)n(])h(w)n(e)f(c) n(ho)r(ose,)g(among)f(the)i(solutions)f(of)h(this)g(equation,)f(a)g(v)n (ery)f(concrete)h(one:)928 173 y Fq(U)985 185 y Fl(k)1025 173 y Fs(\()p Fq(S;)14 b(E)5 b(;)14 b(I)7 b Fs(\))24 b(=)1444 117 y(1)p 1444 154 42 4 v 1447 230 a Fq(b)1510 60 y Fj(Z)1593 81 y Fl(S)1556 249 y(ia)1615 258 y Ff(k)1669 173 y Fq(R)1732 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2131 5404 636 4 v 2131 5510 a Fs(min)2283 5418 y Fj(\020)2336 5510 y Fs(^)-45 b Fq(\024)5 b Fs(^)-48 b Fq(\021)t(;)18 b Fs(^)-46 b Fq(\033)9 b Fs(^)-48 b Fq(\021)s(;)14 b(e)8 b Fs(^)-50 b Fq(\032)2682 5488 y Fs(^)2675 5510 y Fq(\020)2717 5418 y Fj(\021)2777 5424 y Fq(;)-260 5740 y Fs(and)28 b(using)f(\(100\))g(w)n(e)g(obtain)g(b)r (ound)h(\(96\))f(for)1274 5670 y Fj(\015)1274 5719 y(\015)1321 5740 y Fq(R)1385 5710 y Fk(\()o(1\))1473 5670 y Fj(\015)1473 5719 y(\015)1520 5773 y Fk(2)1557 5740 y Fs(.)2350 b Fd(2)1825 5989 y Fs(34)p eop %%Page: 35 35 35 34 bop -260 -82 a Fo(Pr)l(o)l(of)24 b(of)f(The)l(or)l(em)g(4)g (\(\015ow-b)l(ox)f(the)l(or)l(em\).)75 b Fs(Recall)19 b(that,)j(in)d(this)h(app)r(endix,)i(the)e(initial)g(widths)g(in)g(the) g(statemen)n(t)f(of)h(Theorem)f(4,)-260 17 y Fq(\024)p Fs(,)28 b Fq(\033)s Fs(,)h Fq(\021)-18 29 y Fk(1)19 17 y Fs(,)f Fq(\032)113 29 y Fk(3)151 17 y Fs(,)g Fq(\014)t(\016)s Fs(,)g(are)f(written)i(as)e Fq(\024)p Fs(,)h Fq(\033)s Fs(,)h Fq(\021)s Fs(,)f 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1277 y Fl(p)3246 1265 y Fs(=)45 b(^)-48 b Fq(\021)t(=)p Fs(2)3479 1235 y Fl(p)3516 1265 y Fs(,)48 b(^)-50 b Fq(\032)3622 1277 y Fl(p)3700 1265 y Fs(=)47 b(^)-49 b Fq(\032=)p Fs(2)3932 1235 y Fl(p)3969 1265 y Fs(,)-252 1355 y(^)-260 1377 y Fq(\020)-224 1389 y Fl(p)-161 1377 y Fs(=)-65 1355 y(^)-72 1377 y Fq(\020)6 b(=)p Fs(2)54 1347 y Fl(p)91 1377 y Fs(.)39 b(W)-7 b(e)29 b(write)f Fq(\024)558 1389 y Fl(p)620 1377 y Fs(=)c Fq(\024)19 b Fp(\000)f Fs(\()s(^)-45 b Fq(\024)939 1389 y Fk(1)995 1377 y Fs(+)18 b Fp(\001)c(\001)g(\001)k Fs(+)j(^)-45 b Fq(\024)1324 1389 y Fl(p)1362 1377 y Fs(\))q(,)28 b(whic)n(h)g(tends)h(to)f Fq(\024)19 b Fp(\000)i Fs(^)-45 b Fq(\024)p Fs(,)28 b(and)g(analogously)e Fq(\033)2918 1389 y Fl(p)2957 1377 y Fs(,)j Fq(\021)3050 1389 y Fl(p)3088 1377 y Fs(,)g Fq(\032)3183 1389 y Fl(p)3221 1377 y Fs(,)g Fq(\020)3309 1389 y Fl(p)3347 1377 y Fs(.)39 b(W)-7 b(e)29 b(consider)e(the)-260 1476 y(domains)32 b Fp(B)128 1488 y Fl(p)198 1476 y Fs(=)g Fp(B)350 1488 y Fl(\024)389 1496 y Ff(p)423 1488 y Fl(;\033)481 1496 y Ff(p)516 1488 y Fl(;\021)570 1496 y Ff(p)605 1488 y Fl(;\032)659 1496 y Ff(p)694 1488 y Fl(;\020)744 1496 y Ff(p)783 1476 y Fs(,)i(and)f(denote)g Fp(k)o(\001k)1386 1501 y Fl(p)1457 1476 y Fs(and)g Fp(j\001j)1693 1501 y Fl(p)1764 1476 y Fs(the)g(norms)g(on)f Fp(B)2343 1488 y Fl(p)2414 1476 y Fs(\(see)h(Section)g(1.5\).)52 b(Besides,)33 b(w)n(e)g(consider)f (the)-260 1576 y(\014nal)c(domain)f Fp(B)275 1588 y Fi(1)368 1576 y Fs(=)22 b Fp(B)510 1603 y Fl(\024)p Fi(\000)s Fk(^)-36 b Fl(\024;\033)r Fi(\000)t Fk(^)f Fl(\033)r(;\021)r Fi(\000)5 b Fk(^)-38 b Fl(\021)r(;\032)p Fi(\000)6 b Fk(^)-39 b Fl(\032)q(;\020)t Fi(\000)1189 1588 y Fk(^)1183 1603 y Fl(\020)1248 1576 y Fs(and)28 b(the)g(norms)f Fp(k)o(\001k)1909 1601 y Fi(1)2007 1576 y Fs(and)g Fp(j\001j)2237 1601 y Fi(1)2308 1576 y Fs(.)-135 1796 y(In)21 b(order)e(to)i(describ)r (e)f(the)h Fq(p)p Fs(-th)g(iteration,)h(w)n(e)e(start)g(with)i Fq(K)1808 1766 y Fk(\()o Fl(p)p Fi(\000)p Fk(1\))2005 1796 y Fs(=)h Fq(K)2164 1808 y Fk(0)2205 1796 y Fs(+)5 b Fq(Q)2341 1766 y Fk(\()o Fl(p)p Fi(\000)p Fk(1\))2537 1796 y Fs(on)20 b Fp(B)2700 1808 y Fl(p)p Fi(\000)p Fk(1)2823 1796 y Fs(,)i(and)f(write)f Fq(\026)3278 1808 y Fl(p)p Fi(\000)p Fk(1)3425 1796 y Fs(=)3513 1725 y Fj(\015)3513 1775 y(\015)3559 1796 y Fq(Q)3625 1766 y Fk(\()o Fl(p)p Fi(\000)p Fk(1\))3800 1725 y Fj(\015)3800 1775 y(\015)3846 1829 y Fl(p)p Fi(\000)p Fk(1)3969 1796 y Fs(.)-260 1906 y(The)28 b(iterativ)n(e)f(lemma)g(can)g(b)r(e)h(applied)g(if)1172 2178 y Fq(\026)1222 2190 y Fl(p)p Fi(\000)p Fk(1)1368 2178 y Fp(\024)1466 2092 y Fs(min)1618 2000 y Fj(\020)1671 2092 y Fs(^)-45 b Fq(\024)1716 2104 y Fl(p)1760 2092 y Fs(^)d Fq(\021)1795 2104 y Fl(p)1833 2092 y Fq(;)19 b Fs(^)-47 b Fq(\033)1917 2104 y Fl(p)1962 2092 y Fs(^)f Fq(\021)1997 2104 y Fl(p)2036 2092 y Fq(;)14 b(e)8 b Fs(^)-50 b Fq(\032)2155 2104 y Fl(p)2200 2070 y Fs(^)2193 2092 y Fq(\020)2229 2104 y Fl(p)2267 2000 y Fj(\021)p 1466 2159 852 4 v 1815 2235 a Fs(8)p 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Fq(;)2046 2951 y Fs(1)p 1853 2988 426 4 v 1853 3064 a(min)15 b(\()s(^)-45 b Fq(\024)2086 3076 y Fl(p)2124 3064 y Fq(;)19 b Fs(^)-47 b Fq(\033)2208 3076 y Fl(p)2247 3064 y Fs(\))2289 3007 y Fq(;)2353 2951 y Fs(1)p 2336 2988 75 4 v 2343 3056 a(^)2336 3078 y Fq(\020)2372 3090 y Fl(p)2421 3007 y Fq(;)2487 2951 y Fs(1)p 2467 2988 82 4 v 2475 3064 a(^)d Fq(\032)2510 3076 y Fl(p)2559 2865 y Fj(!)2648 3007 y Fs(=)22 b(2)2777 2973 y Fl(p)2815 3007 y Fq(\026)2865 3019 y Fl(p)p Fi(\000)p Fk(1)2992 3007 y Fj(e)-50 b Fq(q)s(:)-135 3347 y Fs(Let)26 b(us)g(c)n(hec)n(k)f (that)h(\(101\))f(holds)h(for)f(ev)n(ery)g Fq(p)e Fp(\025)f Fs(1,)k(and)g(this)h(implies)f(that)g(all)g(the)g(iterations)f(can)h(b) r(e)g(carried)f(out,)h(under)g(the)-260 3446 y(condition)i(of)f (Theorem)g(4.)36 b(First,)28 b(for)f(the)h(\014rst)f(iteration)g(\()p Fq(p)c Fs(=)g(1\))k(w)n(e)h(need)1615 3633 y Fq(\026)1665 3645 y Fk(0)1725 3633 y Fp(\024)1822 3577 y Fq(q)1859 3589 y Fk(1)p 1822 3614 75 4 v 1839 3690 a Fs(4)1930 3633 y 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